Picard-Fuchs Equations for Relative Periods and Abel-Jacobi Map for Calabi-Yau Hypersurfaces (2009)
Li, Si, Lian, Bong H., Yau, Shing-Tung
We study the variation of relative cohomology for a pair consisting of a smooth projective hypersurface and an algebraic subvariety in it. We construct an inhomogeneous Picard-Fuchs equation by...
Cohomology and Hodge Theory on Symplectic Manifolds: I (2009)
Tseng, Li-Sheng, Yau, Shing-Tung
We introduce new finite-dimensional cohomologies on symplectic manifolds. Each exhibits Lefschetz decomposition and contains a unique harmonic representative within each class. Associated with each...
Liu, Chien-Hao, Yau, Shing-Tung
In this continuation of [L-Y1], [L-L-S-Y], [L-Y2], and [L-Y3] (arXiv:0709.1515 [math.AG], arXiv:0809.2121 [math.AG], arXiv:0901.0342 [math.AG], arXiv:0907.0268 [math.AG]), we study D-branes in a...
Yalin Wang, Xiaotian Yin, Jie Zhang, Xianfeng Gu, Tony F. Chan, M. Thompson, ...
Abstract. By solving the Yamabe equation with the discrete surface Ricci flow method, we can conformally parameterize a multiple boundary surface by a multi-hole disk. The resulting parameterizations...
Conformal Slit Mapping and Its Applications to Brain Surface Parameterization (2009)
Yalin Wang, Tonyf. Chan, Paulm. Thompson, Shing-tung Yau
Abstract. We propose a method that computes a conformal mapping from a multiply connected mesh to the so-called slit domain, which consists of a canonical rectangle or disk in which 3D curved...
Liu, Chien-Hao, Yau, Shing-Tung
In this sequel to [L-Y1], [L-L-S-Y], and [L-Y2] (respectively arXiv:0709.1515 [math.AG], arXiv:0809.2121 [math.AG], and arXiv:0901.0342 [math.AG]), we study a D-brane probe on a conifold from the...
Limit of quasilocal mass at spatial infinity (2009)
We study the limit of quasilocal mass defined in [4] and [5] for a family of spacelike 2-surfaces in spacetime. In particular, we show the limit coincides with the ADM mass at spatial infinity. The...
Liu, Chien-Hao, Yau, Shing-Tung
In this continuation of [L-Y1] and [L-L-S-Y], we explain how the Azumaya structure on D-branes together with a netted categorical quotient construction produces the same resolution of ADE orbifold...
POSITIVITY OF QUASI-LOCAL MASS II (2008)
Chiu-chu Melissa, Shing-tung Yau
A spacetime is a four-manifold with a pseudo-metric of signature (+, +, +, −). A hypersurface or a 2-surface in a spacetime is spacelike if the induced metric is positive definite. A quasi-local...
Global Conformal Surface Parameterization (2008)
L. Kobbelt, P. Schröder, Xianfeng Gu, Shing-tung Yau
We solve the problem of computing global conformal parameterizations for surfaces with nontrivial topologies. The parameterization is global in the sense that it preserves the conformality everywhere...
A new geometric approach to problems in birational geometry (2008)
A classical set of birational invariants of a variety are its spaces of pluricanonical forms and some of their canonically defined subspaces. Each of these vector spaces admits a typical metric...
Constructing balanced metrics on some families of non-Kahler Calabi-Yau threefolds (2008)
Fu, Jixiang, Li, Jun, Yau, Shing-Tung
We construct balanced metrics on the family of non-K\"ahler Calabi-Yau threefolds that are obtained by smoothing after contracting $(-1,-1)$-rational curves on K\"ahler Calabi-Yau threefold. As an...
Li, Si, Liu, Chien-Hao, Song, Ruifang, Yau, Shing-Tung
This is a continuation of our study of the foundations of D-branes from the viewpoint of Grothendieck in the region of the related Wilson's theory-space where "branes" are still branes. In this work,...
Global Conformal Surface Parameterization (2008)
L. Kobbelt, P. Schröder, Xianfeng Gu, Shing-tung Yau
We solve the problem of computing global conformal parameterizations for surfaces with nontrivial topologies. The parameterization is global in the sense that it preserves the conformality everywhere...
Uniform Texture Synthesis and Texture Mapping Using Global Parameterization (2008)
Lujin Wang, Xianfeng Gu, Klaus Mueller, Shing-tung Yau
Abstract Texture mapping and texture synthesis are two popular methods for the decoration of surfaces with visual detail. Here, an existing challenge is to preserve, or at least balance, two...
New Heterotic Non-Kahler Geometries (2008)
Becker, Melanie, Tseng, Li-Sheng, Yau, Shing-Tung
New heterotic torsional geometries are constructed as orbifolds of T^2 bundles over K3. The discrete symmetries considered can be freely-acting or have fixed points and/or fixed curves. We give...
Local Heterotic Torsional Models (2008)
Fu, Ji-Xiang, Tseng, Li-Sheng, Yau, Shing-Tung
We present a class of smooth supersymmetric heterotic solutions with a non-compact Eguchi-Hanson space. The non-compact geometry is embedded as the base of a six-dimensional non-Kahler manifold with...
Isometric embeddings into the Minkowski space and new quasi-local mass (2008)
The definition of quasi-local mass for a bounded space-like region in space-time is essential in several major unsettled problems in general relativity. The quasi-local mass is expected to be a type...
Quasilocal mass in general relativity (2008)
There have been many attempts to define the notion of quasilocal mass for a spacelike 2-surface in spacetime by the Hamilton-Jacobi analysis. The essential difficulty in this approach is to identify...
Geometric compression using Riemann surface structure,Comm. in info. sys (2008)
Xianfeng Gu, Yalin Wang, Shing-tung Yau
Abstract. This paper introduces a theoretic result that shows any surface in 3 dimensional Euclidean space can be determined by its conformal factor and mean curvature uniquely up to rigid motions....
COMPUTING CONFORMAL INVARIANTS: PERIOD MATRICES ∗ (2008)
Xianfeng Gu, Yalin Wang, Shing-tung Yau
Abstract. This work introduces a system of algorithms to compute period matrices for general surfaces with arbitrary topologies. The algorithms are intrinsic to the geometry, and independent of...
CONFORMAL SPHERICAL PARAMETRIZATION FOR HIGH GENUS SURFACES (2008)
Wei Zeng, Xin Li, Shing-tung Yau, Xianfeng Gu
Abstract. Surface parameterization establishes bijective maps from a surface onto a topolog-ically equivalent standard domain. It is well known that the spherical parameterization is limited to...
CONFORMAL SPHERICAL PARAMETRIZATION FOR HIGH GENUS SURFACES (2008)
Wei Zeng, Xin Li, Shing-tung Yau, Xianfeng Gu
Abstract. Surface parameterization establishes bijective maps from a surface onto a topolog-ically equivalent standard domain. It is well known that the spherical parameterization is limited to...
Chinese Academy of Sciences (2008)
Wei Zeng, Xin Li, Shing-tung Yau, Xianfeng Gu
Surface parameterization establishes bijective maps from a surface onto a topologically equivalent standard domain. It is well known that the spherical parameterization is limited to genus-zero...
ACCEPTED MANUSCRIPT Geometric Accuracy Analysis for Discrete Surface Approximation (2008)
Junfei Dai, Wei Luo, Miao Jin, Wei Zeng, Ying He, Shing-tung Yau, ...
doi: 10.1016/j.cagd.2007.04.004 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the...
SURFACE SEGMENTATION USING GLOBAL CONFORMAL STRUCTURE ∗ (2008)
Yalin Wang, Xianfeng Gu, Shing-tung Yau
Abstract. Surface segmentation is a fundamental problem in computer graphics. It has various applications such as metamorphosis, surface matching, surface compression, 3D shape retrieval, texture...
Uniform Texture Synthesis and Texture Mapping Using Global Parameterization (2008)
Lujin Wang, Xianfeng Gu, Klaus Mueller, Shing-tung Yau
Abstract Texture mapping and texture synthesis are two popular methods for the decoration of surfaces with visual detail. Here, an existing challenge is to preserve, or at least balance, two...
Local geometry of the G2 moduli space (2008)
Grigorian, Sergey, Yau, Shing-Tung
We consider deformations of torsion-free G2 structures, defined by the G2-invariant 3-form $\phi$ and compute the expansion of the Hodge star of $\phi$ to fourth order in the deformations of $\phi$....
Linear Waves in the Kerr Geometry: A Mathematical Voyage to Black Hole Physics (2008)
Finster, Felix, Kamran, Niky, Smoller, Joel, Yau, Shing-Tung
This paper gives a survey of wave dynamics in the Kerr space-time geometry, the mathematical model of a rotating black hole in equilibrium. After a brief introduction to the Kerr metric, we review...
Taming symplectic forms and the Calabi-Yau equation (2008)
Tosatti, Valentino, Weinkove, Ben, Yau, Shing-Tung
We study the Calabi–Yau equation on symplectic manifolds. We show that Donaldson's conjecture on estimates for this equation in terms of a taming symplectic form can be reduced to an integral...
Fourier-Mukai Transform and Mirror Symmetry for D-Branes on Elliptic Calabi-Yau (2007)
Gottfried Curio, Shing-tung Yau
Fibrewise T-duality (Fourier-Mukai transform) for D-branes on an elliptic Calabi-Yau three-fold X is seen to have an expected adiabatic form for its induced cohomology operation only when an...
Liu, Chien-Hao, Yau, Shing-Tung
We explain how Polchinski's work on D-branes re-read from a noncommutative version of Grothendieck's equivalence of local geometries and function rings gives rise to an intrinsic prototype definition...
Heterotic Kahler/non-Kahler Transitions (2007)
Becker, Melanie, Tseng, Li-Sheng, Yau, Shing-Tung
We show how two topologically distinct spaces - the Kahler K3 x T^2 and the non-Kahler T^2 bundle over K3 - can be smoothly connected in heterotic string theory. The transition occurs when the base...
Taming symplectic forms and the Calabi-Yau equation (2007)
Tosatti, Valentino, Weinkove, Ben, Yau, Shing-Tung
We study the Calabi-Yau equation on symplectic manifolds. We show that Donaldson's conjecture on estimates for this equation in terms of a taming symplectic form can be reduced to an integral...
A Rigorous Treatment of Energy Extraction from a Rotating Black Hole (2007)
Finster, Felix, Kamran, Niky, Smoller, Joel, Yau, Shing-Tung
The Cauchy problem is considered for the scalar wave equation in the Kerr geometry. We prove that by choosing a suitable wave packet as initial data, one can extract energy from the black hole,...
Brain surface conformal parameterization using riemann surface structure (2007)
Yalin Wang, Lok Ming Lui, Xianfeng Gu, Kiralee M. Hayashi, Tony F. Chan, Arthur W. Toga, ...
Abstract—In medical imaging, parameterized 3-D surface models are useful for anatomical modeling and visualization, statistical comparisons of anatomy, and surface-based registration and signal...
Moduli Space of Torsional Manifolds (2006)
Becker, Melanie, Tseng, Li-Sheng, Yau, Shing-Tung
We characterize the geometric moduli of non-Kaehler manifolds with torsion. Heterotic supersymmetric flux compactifications require that the six-dimensional internal manifold be balanced, the gauge...
Liu, Chien-Hao, Yau, Shing-Tung
We study the degeneration and the gluing of Kuranishi structures in Gromov-Witten theory under a symplectic cut. This leads us to a degeneration axiom and a gluing axiom for open Gromov-Witten...
Structures of Three-Manifilds (2006)
This is lecture notes of a talk I gave at the Morningside Center of Mathematics on June 20, 2006. In this talk, I survey on Poincare and geometrization conjecture.
Obstructions to the Existence of Sasaki-Einstein Metrics (2006)
Gauntlett, Jerome P., Martelli, Dario, Sparks, James, Yau, Shing-Tung
We describe two simple obstructions to the existence of Ricci-flat Kahler cone metrics on isolated Gorenstein singularities or, equivalently, to the existence of Sasaki-Einstein metrics on the links...
Decay of Solutions of the Wave Equation in the Kerr Geometry (2006)
Yau, Shing-Tung, Finster, Felix, Kamran, N., Smoller, Joel A.
We consider the Cauchy problem for the massless scalar wave equation in the Kerr geometry for smooth initial data compactly supported outside the event horizon. We prove that the solutions decay in...
Branes, Bundles and Attractors: Bogomolov and Beyond (2006)
Douglas, Michael R., Reinbacher, Rene, Yau, Shing-Tung
We discuss conjectures following from the attractor mechanism in type II string theory about the possible Chern classes of stable holomorphic vector bundles on Calabi-Yau threefolds. In particular,...
Anomaly Cancellation and Smooth Non-Kahler Solutions in Heterotic String Theory (2006)
Becker, Katrin, Becker, Melanie, Fu, Ji-Xiang, Tseng, Li-Sheng, Yau, Shing-Tung
We show that six-dimensional backgrounds that are T^2 bundle over a Calabi-Yau two-fold base are consistent smooth solutions of heterotic flux compactifications. We emphasize the importance of the...
The purpose of this paper is to solve a problem posed by Strominger in constructing smooth models of superstring theory with flux. These are given by non-Kahler manifolds with torsion.
Sasaki-Einstein Manifolds and Volume Minimisation (2006)
Martelli, Dario, Sparks, James, Yau, Shing-Tung
We study a variational problem whose critical point determines the Reeb vector field for a Sasaki-Einstein manifold. This extends our previous work on Sasakian geometry by lifting the condition that...
Perspectives on geometric analysis (2006)
This is a survey paper on several aspects of differential geometry for the last 30 years, especially in those areas related to non-linear analysis. It grew from a talk I gave on the occasion of...
A generalization of Liu-Yau's quasi-local mass (2006)
In \cite{ly, ly2}, Liu and the second author propose a definition of the quasi-local mass and prove its positivity. This is demonstrated through an inequality which in turn can be interpreted as a...
Brain surface conformal parameterization with algebraic functions (2006)
Yalin Wang, Xianfeng Gu, Tony F. Chan, Paul M. Thompson, Shing-tung Yau
In medical imaging, parameterized 3D surface models are of great interest for anatomical modeling and visualization, statistical comparisons of anatomy, and surface-based registration and signal...
Geometric Accuracy Analysis for Discrete Surface Approximation (2006)
Junfei Dai, Wei Luo, Shing-tung Yau, Xianfeng David Gu
In geometric modeling and processing, computer graphics, smooth surfaces are approximated by discrete triangular meshes reconstructed from sample points on the surface. A fundamental problem is to...
Brain surface conformal parameterization with algebraic functions (2006)
Yalin Wang, Xianfeng Gu, Tony F. Chan, Paul M. Thompson, Shing-tung Yau
Abstract. In medical imaging, parameterized 3D surface models are of great interest for anatomical modeling and visualization, statistical comparisons of anatomy, and surface-based registration and...
Gu, Xianfeng, Zhang, Song, Huang, Peisen, Zhang, Liangjun, Yau, Shing-Tung, Martin, Ralph Robert
We introduce a novel geometric representation called a holoimage, which encodes both shading and geometry information within the same image, based on the principles of wave optics. ‘Image’...
Minimization with the affine normal direction (2005)
Cheng, Hsiao-Bing, Cheng, Li-Tien, Yau, Shing-Tung
In this paper, we consider minimization of a real-valued function $f$ over $\bold R\sp {n+1}$ and study the choice of the affine normal of the level set hypersurfaces of $f$ as a direction for...
Yau, Shing-Tung, Finster, Felix, Kamran, N., Smoller, Joel A.
We consider the scalar wave equation in the Kerr geometry for Cauchy data which is smooth and compactly supported outside the event horizon. We derive an integral representation which expresses the...
Existence of Supersymmetric Hermitian Metrics with Torsion on Non-Kaehler Manifolds (2005)
We proved the existence of supersymmetric Hermitian metrics with torsion on a class of non-Kaehler manifolds.
Affine manifolds, SYZ geometry and the "Y" vertex (2005)
Loftin, John, Yau, Shing-Tung, Zaslow, Eric
We prove the existence of a solution to the Monge-Ampère equation detHess(ø) = 1 on a cone over a thrice-punctured two-sphere. The total space of the tangent bundle is thereby a Calabi-Yau manifold...
Liu, Chien-Hao, Yau, Shing-Tung
We study how Gromov-Witten invariants of projective 3-folds transform under a standard flop and a small extremal transition in the algebro-geometric setting from the recent development of algebraic...
Decay of Solutions of the Wave Equation in the Kerr Geometry (2005)
Finster, Felix, Kamran, Niky, Smoller, Joel, Yau, Shing-Tung
We consider the Cauchy problem for the scalar wave equation in the Kerr geometry for smooth initial data supported outside the event horizon. We prove that the solutions decay in time in...
The Geometric Dual of a-maximisation for Toric Sasaki-Einstein Manifolds (2005)
Martelli, Dario, Sparks, James, Yau, Shing-Tung
We show that the Reeb vector, and hence in particular the volume, of a Sasaki-Einstein metric on the base of a toric Calabi-Yau cone of complex dimension n may be computed by minimising a function Z...
Brain surface parameterization using Riemann Surface structure (2005)
Yalin Wang, Kiraleem. Hayashi, Tonyf. Chan, Paul M. Thompson, Shing-tung Yau
Abstract. We develop a general approach that uses holomorphic 1forms to parameterize anatomical surfaces with complex (possibly branching) topology. Rather than evolve the surface geometry to a plane...
Surface parameterization using riemann surface structure (2005)
Yalin Wang, Tony F. Chan, Xianfeng Gu, Paul M. Thompson, Kiralee M. Hayashi, Shing-tung Yau
We propose a general method that parameterizes general surfaces with complex (possible branching) topology using Riemann surface structure. Rather than evolve the surface geometry to a plane or...
Surface parameterization using riemann surface structure (2005)
Yalin Wang, Tony F. Chan, Xianfeng Gu, Paul M. Thompson, Kiralee M. Hayashi, Shing-tung Yau
We propose a general method that parameterizes general surfaces with complex (possible branching) topology using Riemann surface structure. Rather than evolve the surface geometry to a plane or...
Positivity of quasi-local mass II (2004)
Liu, Chiu-Chu Melissa, Yau, Shing-Tung
We prove the following stronger verson of the positivity of quasi-local mass stated in gr-qc/0303019: the quasi-local energy (mass) of each connected component of the boundary of a compact spacelike...
The Existence of Supersymmetric String Theory with Torsion (2004)
We derived an existence criterion to the Supersymmetric String Theory with Torsion proposed by Strominger and proved the existence of such theory for a class of Calabi-Yau threefolds.
Liu, Chien-Hao, Yau, Shing-Tung
The study of open/closed string duality and large $N$ duality suggests a Gromov-Witten theory for conifolds that sits on the border of both a closed Gromov-Witten theory and an open Gromov-Witten...
Canonical Metrics on the Moduli Space of Riemann Surfaces II (2004)
Liu, Kefeng, Sun, Xiaofeng, Yau, Shing-Tung
In this paper we continue our study on the canonical metrics on the Teichm\"uller and the moduli space of Riemman surfaces. We first prove the equivalence of the Bergman metric and the Carath\'eodory...
Liu, Chien-Hao, Yau, Shing-Tung
In two very detailed, technical, and fundamental works, Jun Li constructed a theory of Gromov-Witten invariants for a singular scheme of the gluing form $Y_1\cup_D Y_2$ that arises from a...
Topological String Partition Functions as Polynomials (2004)
Yamaguchi, Satoshi, Yau, Shing-Tung
We investigate the structure of the higher genus topological string amplitudes on the quintic hypersurface. It is shown that the partition functions of the higher genus than one can be expressed as...
Affine Manifolds, SYZ Geometry, and the "Y" Vertex (2004)
Loftin, John, Yau, Shing-Tung, Zaslow, Eric
We study the real Monge-Amp\`ere equation in two and three dimensions, both from the point of view of the SYZ conjecture, where solutions give rise to semi-flat Calabi-Yau's and in affine...
Canonical Metrics on the Moduli Space of Riemann Surfaces I (2004)
Liu, Kefeng, Sun, Xiaofeng, Yau, Shing-Tung
We prove the equivalences of several classical complete metrics on the Teichm\"uller and the moduli spaces of Riemann surfaces. We use as bridge two new K\"ahler metrics, the Ricci metric and the...
Liu, Chien-Hao, Liu, Kefeng, Yau, Shing-Tung
In [L-L-Y1, III: Sec. 5.4] on mirror principle, a method was developed to compute the integral $\int_{X}\tau^{\ast}e^{H\cdot t}\cap {\mathbf 1}_d$ for a flag manifold $X=\Fl_{r_1, ..., r_I}({\Bbb...
On A-twisted Moduli Stack for Curves from Witten's Gauged Linear Sigma Models (2004)
Liu , Chien-Hao, Liu , Kefeng, Yau , Shing-Tung
Witten's gauged linear sigma model [Wi1] is one of the universal frameworks or structures that lie behind stringy dualities. Its Atwisted moduli space at genus 0 case has been used in the Mirror...
Surface Segmentation Using Global Conformal Structure (2004)
Gu, Xianfeng, Wang, Yalin, Yau, Shing-Tung
Surface segmentation is a fundamental problem in computer graphics. It has various applications such as metamorphosis, surface matching, surface compression, 3D shape retrieval, texture mapping, etc....
Genus zero surface conformal mapping and its application to brain surface mapping (2004)
Xianfeng Gu, Yalin Wang, Tony F. Chan, Paul M. Thompson, Shing-tung Yau
Abstract—We developed a general method for global conformal parameterizations based on the structure of the cohomology group of holomorphic one-forms for surfaces with or without boundaries (Gu and...
Genus zero surface conformal mapping and its application to brain surface mapping (2004)
Xianfeng Gu, Yalin Wang, Tony F. Chan, Paul M. Thompson, Shing-tung Yau
Abstract. It is well known that any genus zero surface can be mapped conformally onto the sphere and any local portion thereof onto a disk. However, it is not trivial to find a general method which...
Genus zero surface conformal mapping and its application to brain surface mapping (2004)
Xianfeng Gu, Yalin Wang, Tony F. Chan, Paul M. Thompson, Shing-tung Yau
Abstract. It is well known that any genus zero surface can be mapped conformally onto the sphere and any local portion thereof onto a disk. However, it is not trivial to find a general method which...
Intrinsic brain surface conformal mapping using a variational method (2004)
Yalin Wang, Xianfeng Gu, Tony F. Chan, Paul M. Thompson, Shing-tung Yau
Abstract. We propose a new variational method which can find a unique conformal mapping between any two genus zero manifolds by minimizing the harmonic energy of the map and apply it to the cortical...
Genus zero surface conformal mapping and its application to brain surface mapping (2004)
Xianfeng Gu, Yalin Wang, Tony F. Chan, Paul M. Thompson, Shing-tung Yau
It is well known that any genus zero surface can be mapped conformally onto the sphere and any local portion thereof onto a disk. However, it is not trivial to find a general method which finds a...
Genus Zero Surface Conformal Mapping and Its Application to Brain Surface Mapping (2004)
Xianfeng Gu, Yalin Wang, Tony F. Chan, Paul M. Thompson, Shing-tung Yau
It is well known that any genus zero surface can be mapped conformally onto the sphere and any local portion thereof onto a disk. However, it is not trivial to find a general method which computes a...
Genus Zero Surface Conformal Mapping and Its Application to Brain Surface Mapping (2004)
Xianfeng Gu, Yalin Wang, Tony F. Chan, Paul M. Thompson, Shing-tung Yau
We developed a general method for global conformal parameterizations based on the structure of the cohomology group of holomorphic one-forms with or without boundaries [1], [2]. For genus zero...
Volumetric Harmonic Brain Mapping (2004)
Yalin Wang Xianfeng, Yalin Wang, Xianfeng Gu, Tony F. Chan, Paul M. Thompson, Shing-tung Yau
In [1], we developed two different techniques to study volume mapping problem in Computer Graphics. The first one is to find a harmonic map from a 3 manifold to a 3D solid sphere and the second is a...
Yalin Wang, Xianfeng Gu, Paul M. Thompson, Shing-tung Yau
Abstract. We developed two techniques to address 3D volume parameterization and deformation mapping problems that arise in medical imaging [1]. The first algorithm finds a harmonic map from a...
Optimal global conformal surface parameterization (2004)
Miao Jin, Yalin Wang, Shing-tung Yau, Xianfeng Gu
Figure 1: Uniform global conformal parameterization ((a) and (b)) and region emphasized conformal parameterization ((c) and (d)). (a). Least uniform conformal parameterization with energy: 21.208e...
Optimal global conformal surface parameterization (2004)
Miao Jin, Yalin Wang, Shing-tung Yau, Xianfeng Gu
Figure 1: Uniform global conformal parameterization ((a) and (b)) and region emphasized conformal parameterization ((c) and (d)). (a). Least uniform conformal parameterization with energy: 21.208e...
Optimal global conformal surface parameterization (2004)
Visualization Miao Jin, Yalin Wang, Xianfeng Gu, Shing-tung Yau
Abstract. All orientable metric surfaces are Riemann surfaces and admit global conformal parameterizations. Riemann surface structure is a fundamental structure and governs many natural physical...
Kummer structures on a K3 surface: An old question of T. Shioda (2003)
Hosono, Shinobu, Lian, Bong H., Oguiso, Keiji, Yau, Shing-Tung
We apply our earlier results on Fourier-Mukai partners to answer definitively a question about Kummer surface structures posed by T. Shioda twenty-five years ago.
Finster, Felix, Kamran, Niky, Smoller, Joel, Yau, Shing-Tung
We consider the scalar wave equation in the Kerr geometry for Cauchy data which is smooth and compactly supported outside the event horizon. We derive an integral representation which expresses the...
Shafarevich's Conjecture for CY Manifolds I (2003)
Liu, Kefeng, Todorov, Andrey, Yau, Shing-Tung, Zuo, Kang
In this paper we first study the moduli spaces related to Calabi-Yau manifolds. We then apply the results to the following problem. Let $C$ be a fixed Riemann surface with fixed finite number of...
Positivity of Quasilocal Mass (2003)
Liu, Chiu-Chu Melissa, Yau, Shing-Tung
Motivated by the important work of Brown adn York on quasilocal energy, we propose definitions of quasilocal energy and momentum surface energy of a spacelike 2-surface with positive intrinsic...
The long-time dynamics of Dirac particles in the Kerr-Newman black hole geometry (2003)
Finster, Felix, Kamran, Niky, Smoller, Joel, Yau, Shing-Tung
We consider the Cauchy problem for the massive Dirac equation in the non-extreme Kerr-Newman geometry outside the event horizon. We derive an integral representation for the Dirac propagator...
Counting Unimodular Lattices in $\R^{r,s}$ (2003)
Hosono, Shinobu, Lian, Bong H., Oguiso, Keiji, Yau, Shing-Tung
Narain lattices are unimodular lattices {\it in} $\R^{r,s}$, subject to certain natural equivalence relation and rationality condition. The problem of describing and counting these rational...
Computing Conformal Invariants: Period Matrices (2003)
Gu , Xianfeng, Wang , Yalin, Yau , Shing-Tung
This work introduces a system of algorithms to compute period matrices for general surfaces with arbitrary topologies. The algorithms are intrinsic to the geometry, and independent of surface...
Geometric Compression Using Riemann Surface Structure (2003)
Gu , Xianfeng, Wang , Yalin, Yau , Shing-Tung
This paper introduces a theoretic result that shows any surface in 3 dimensional Euclidean space can be determined by its conformal factor and mean curvature uniquely up to rigid motions. This...
Volumetric Harmonic Map (2003)
Gu , Xianfeng, Wang , Yalin, Yau , Shing-Tung
We develop two different techniques to study volume mapping problem in Computer Graphics and Medical Imaging fields. The first one is to find a harmonic map from a 3 manifold to a 3D solid sphere and...
Computing Conformal Invariants: Period Matrices (2003)
Xianfeng Gu, Yalin Wang, Shing-tung Yau
This work introduces a system of algorithms to compute period matrices for general surfaces with arbitrary topologies. The algorithms are intrinsic to the geometry, independent of surface...
Surface Classification Using Conformal Structures (2003)
D surface classification is a fundamental problem in computer vision and computational geometry. Surfaces can be classified by different transformation groups. Traditional classification methods...
Surface classification using conformal structures (2003)
3D surface classification is a fundamental problem in computer vision and computational geometry. Surfaces can be classified by different transformation groups. Traditional classification methods...
Intrinsic Brain Surface Conformal Mapping (2003)
Using Variational Method, Yalin Wang, Xianfeng Gu, Tony F. Chan, Paul M. Thompson, Shing-tung Yau
We propose a new variational method which can find a unique conformal mapping between any two genus zero manifolds by minimizing the harmonic energy of the map and apply it to the cortical surface...
On A-twisted moduli stack for curves from Witten's gauged linear sigma models (2002)
Liu, Chien-Hao, Liu, Kefeng, Yau, Shing-Tung
Witten's gauged linear sigma model [Wi1] is one of the universal frameworks or structures that lie behind stringy dualities. Its A-twisted moduli space at genus 0 case has been used in the Mirror...
Computing Conformal Structure of Surfaces (2002)
This paper solves the problem of computing conformal structures of general 2-manifolds represented as triangle meshes. We compute conformal structures in the following way: first compute homology...
c=2 Rational Toroidal Conformal Field Theories via the Gauss Product (2002)
Hosono, Shinobu, Lian, Bong H., Oguiso, Keiji, Yau, Shing-Tung
We find a concise relation between the moduli $\tau, \rho$ of a rational Narain lattice $\Gamma(\tau,\rho)$ and the corresponding momentum lattices of left and right chiral algebras via the Gauss...
Fourier-Mukai partners of a K3 surface of Picard number one (2002)
Hosono, Shinobu, Lian, Bong H., Oguiso, Keiji, Yau, Shing-Tung
We shall give a complete geometrical description of the FM partners of a K3 surface of Picard number 1 and its applications.
Finster, Felix, Smoller, Joel, Yau, Shing-Tung
This short note compares different methods to prove that Einstein-Dirac systems have no static, spherically symmetric solutions.
Smoller, Joel A., Kamran, N., Finster, Felix, Yau, Shing-Tung
The Cauchy problem is considered for the massive Dirac equation in the non-extreme Kerr–Newman geometry, for smooth initial data with compact support outside the event horizon and bounded angular...
Duality and Fibrations on G_2 Manifolds (2002)
Gukov, Sergei, Yau, Shing-Tung, Zaslow, Eric
We argue that G_2 manifolds for M-theory admitting string theory Calabi-Yau duals are fibered by coassociative submanifolds. Dual theories are constructed using the moduli space of M5-brane fibers as...
Kummer structures on a K3 surface - An old question of T. Shioda (2002)
Hosono, Shinobu, Lian, Bong H., Oguiso, Keiji, Yau, Shing-Tung
We apply our earlier results on Fourier-Mukai partners to answer definitively a question about Kummer surface structures, posed by T. Shioda 25 years ago.
Fourier-Mukai number of a K3 surface (2002)
Hosono, Shinobu, Lian, Bong H., Oguiso, Keiji, Yau, Shing-Tung
We shall give a Counting Formula for the number of Fourier-Mukai partners of a K3 surface and consider three applications.
Autoequivalences of Derived Category of A K3 Surface and Monodromy Transformations (2002)
Hosono, Shinobu, Lian, Bong H., Oguiso, Keiji, Yau, Shing-Tung
We consider autoequivalences of the bounded derived category of coherent sheaves on a K3 surface. We prove that the image of the autoequivalences has index at most two in the group of the Hodge...
Computing conformal structures of surfaces (2002)
Abstract. This paper solves the problem of computing conformal structures of general 2manifolds represented as triangular meshes. We approximate the De Rham cohomology by simplicial cohomology and...
Computing Conformal Structures Of Surfaces (2002)
This paper solves the problem of computing conformal structures of general 2manifolds represented as triangular meshes. We approximate the De Rham cohomology by simplicial cohomology and represent...
The $S^1$ fixed points in Quot-schemes and mirror principle computations (2001)
Lian, Bong H., Liu, Chien-Hao, Liu, Kefeng, Yau, Shing-Tung
We describe the $S^1$-action on the Quot-scheme $\Quot({\cal E}^n)$ associated to the trivial bundle ${\cal E}^n=CP^1\times{\smallBbb C}^n$. In particlular, the topology of the $S^1$-fixed-point...
Geometry of three manifolds and existence of Black Hole due to boundary effect (2001)
In this paper, we observe that the brane functional studied in hep-th/9910245 can be used to generalize some of the works that Schoen and I [4] did many years ago. The key idea is that if a three...
Finster, Felix, Kamran, Niky, Smoller, Joel, Yau, Shing-Tung
The Cauchy problem is considered for the massive Dirac equation in the non-extreme Kerr-Newman geometry, for smooth initial data with compact support outside the event horizon and bounded angular...
Fibrewise T-Duality for D-Branes on Elliptic Calabi-Yau (2001)
Andreas, Bjorn, Curio, Gottfried, Ruiperez, Daniel Hernandez, Yau, Shing-Tung
Fibrewise T-duality (Fourier-Mukai transform) for D-branes on an elliptic Calabi-Yau $X$ is shown to require naturally an appropriate twisting of the operation respectively a twisted charge. The...
Towards a Mirror Principle for Higher Genus. Joint with (2001)
Bong Lian, Kefeng Liu, Shing-tung Yau
Mirror principle is a general method developed in [LLY1]-[LLY4] to compute characteristic classes and characteristic numbers on moduli spaces of stable maps in terms of hypergeometric type series....
Fourier-Mukai Transform and Mirror Symmetry for D-Branes on Elliptic Calabi-Yau (2000)
Andreas, Bjorn, Curio, Gottfried, Ruiperez, Daniel Hernandez, Yau, Shing-Tung
Fibrewise T-duality (Fourier-Mukai transform) for D-branes on an elliptic Calabi-Yau three-fold $X$ is seen to have an expected adiabatic form for its induced cohomology operation only when an...
Toric morphisms and fibrations of toric Calabi-Yau hypersurfaces (2000)
Hu, Yi, Liu, Chien-Hao, Yau, Shing-Tung
Special fibrations of toric varieties have been used by physicists, e.g. the school of Candelas, to construct dual pairs in the study of Het/F-theory duality. Motivated by this, we investigate in...
A Survey of Mirror Principle (2000)
Lian, Bong H., Liu, Kefeng, Yau, Shing-Tung
This note briefly reviews the {\it Mirror Principle} as developed in the series of papers \LLYI\LLYII\LLYIII\LLYIV\LCHY. We illustrate this theory with a few new examples. One of them gives an...
Maximal Unipotent Monodromy for Complete Intersection CY Manifolds (2000)
Lian, Bong H., Todorov, Andrey, Yau, Shing-Tung
The computations that are suggested by String Theory in the B model requires the existence of degenerations of CY manifolds with maximum unipotent monodromy. In String Theory such a point in the...
Lian, Bong H., Liu, Kefeng, Yau, Shing-Tung
This is a continuation of "Mirror Principle III"(math.AG/9912038).
The Long-Time Dynamics of Dirac Particles in the Kerr-Newman Black Hole Geometry (2000)
Finster, Felix, Kamran, Niky, Smoller, Joel, Yau, Shing-Tung
We consider the Cauchy problem for the massive Dirac equation in the non-extreme Kerr-Newman geometry outside the event horizon. We derive an integral representation for the Dirac propagator...
From Special Lagrangian to Hermitian-Yang-Mills via Fourier-Mukai Transform (2000)
Leung, Naichung Conan, Yau, Shing-Tung, Zaslow, Eric
We exhibit a transformation taking special Lagrangian submanifolds of a Calabi-Yau together with local systems to vector bundles over the mirror manifold with connections obeying deformed...
Finster, Felix, Smoller, Joel, Yau, Shing-Tung
We study a static, spherically symmetric system of (2j+1) massive Dirac particles, each having angular momentum j, j=1,2,..., in a classical gravitational and SU(2) Yang-Mills field. We show that for...
A reconstruction of Euler data (2000)
Lian, Bong H., Liu, Chien-Hao, Yau, Shing-Tung
We apply the mirror principle of [L-L-Y] to reconstruct the Euler data $Q=\{Q_d\}_{d\in{\tinyBbb N}\cup\{0\}}$ associated to a vector bundle $V$ on ${\smallBbb C}{\rm P}^n$ and a multiplicative class...
On the Splitting Type of an Equivariant Vector Bundle over a Toric Manifold (2000)
Liu, Chien-Hao, Yau, Shing-Tung
From the work of Lian, Liu, and Yau on "Mirror Principle", in the explicit computation of the Euler data $Q=\{Q_0, Q_1, ... \}$ for an equivariant concavex bundle ${\cal E}$ over a toric manifold,...
Some Recent Progress in Classical General Relativity (2000)
Finster, Felix, Smoller, Joel, Yau, Shing-Tung
In this short survey paper, we discuss certain recent results in classical gravity. Our main attention is restricted to two topics: the positive mass conjecture and its extensions to the case with...
The Interaction of Dirac Particles with Non-Abelian Gauge Fields and Gravity - Bound States (2000)
Finster, Felix, Smoller, Joel, Yau, Shing-Tung
We consider a spherically symmetric, static system of a Dirac particle interacting with classical gravity and an SU(2) Yang-Mills field. The corresponding Einstein-Dirac-Yang/Mills equations are...
From Special Lagrangian to Hermitian-Yang-Mills via Fourier-Mukai Transform (2000)
Naichung Conan Leung, Shing-tung Yau, Eric Zaslow
We exhibit a transformation taking special Lagrangian submanifolds of a Calabi-Yau together with local systems to vector bundles over the mirror manifold with connections obeying deformed...
Naichung Conan Leung, Shing-tung Yau, Eric Zaslow, Supersymmetric A
We exhibit a transformation taking special Lagrangian submanifolds of a Calabi-Yau together with local systems to vector bundles over the mirror manifold with connections obeying deformed...
The Interaction of Dirac Particles with Non-Abelian Gauge Fields and Gravity - Black Holes (1999)
Finster, Felix, Smoller, Joel, Yau, Shing-Tung
We consider a static, spherically symmetric system of a Dirac particle in a classical gravitational and SU(2) Yang-Mills field. We prove that the only black-hole solutions of the corresponding...
The Einstein-Dirac-Maxwell Equations - Black Hole Solutions (1999)
Finster, Felix, Smoller, Joel, Yau, Shing-Tung
In this summary article, we review and discuss the non-existence of stationary black hole solutions for the Einstein-Dirac-Maxwell equations.
Yau, Shing-Tung, Smoller, Joel A., Finster, Felix
We consider for j =½, … a spherically symmetric, static system of (2 j +1) Dirac particles, each having total angular momentum j . The Dirac particles interact via a classical gravitational and...
The Coupling of Gravity to Spin and Electromagnetism (1999)
Finster, Felix, Smoller, Joel, Yau, Shing-Tung
The coupled Einstein-Dirac-Maxwell equations are considered for a static, spherically symmetric system of two fermions in a singlet spinor state. Stable soliton-like solutions are shown to exist, and...
Finster, Felix, Kamran, Niky, Smoller, Joel, Yau, Shing-Tung
We prove that, in the non-extreme Kerr-Newman black hole geometry, the Dirac equation has no normalizable, time-periodic solutions. A key tool is Chandrasekhar's separation of the Dirac equation in...
Regularity of harmonic maps (1999)
Yalin Wang, Xianfeng Gu, Shing-tung Yau
Abstract. We develop two different techniques to study volume mapping problem in Computer Graphics and Medical Imaging fields. The first one is to find a harmonic map from a 3 manifold to a 3D solid...
Finster, Felix, Smoller, Joel, Yau, Shing-Tung
We consider for j=1/2, 3/2,... a spherically symmetric, static system of (2j+1) Dirac particles, each having total angular momentum j. The Dirac particles interact via a classical gravitational and...
Absence of Zero Energy States in the Simplest d=3 (d=5?) Matrix Models (1998)
The method introduced in [hep-th/9805020] is simplified, and used to calculate the asymptotic form of all SU(2) \times SO(d=3, resp. 5) invariant wave functions satisfying $Q_{\hat{\beta}} \Psi = 0,...
Finster, Felix, Smoller, Joel, Yau, Shing-Tung
It is shown analytically that the Dirac equation has no normalizable, time-periodic solutions in a Reissner-Nordstrom black hole background; in particular, there are no static solutions of the Dirac...
Particle-Like Solutions of the Einstein-Dirac-Maxwell Equations (1998)
Finster, Felix, Smoller, Joel, Yau, Shing-Tung
We consider the coupled Einstein-Dirac-Maxwell equations for a static, spherically symmetric system of two fermions in a singlet spinor state. Soliton-like solutions are constructed numerically. The...
Particle-Like Solutions of the Einstein-Dirac Equations (1998)
Finster, Felix, Smoller, Joel, Yau, Shing-Tung
The coupled Einstein-Dirac equations for a static, spherically symmetric system of two fermions in a singlet spinor state are derived. Using numerical methods, we construct an infinite number of...
Shing-tung Yau, Non-existence Black, Hole Solutions Spherically, Felix Finster, Felix Finster, ...
We consider for j = 1 2 ; 3 2 ; : : : a spherically symmetric, static system of (2j + 1) Dirac particles, each having total angular momentum j. The Dirac particles interact via a classical...
Absence of Zero Energy States in Reduced SU(N) 3d Supersymmetric Yang Mills Theory (1997)
For the SU(N) invariant supersymmetric matrix model related to membranes in 4 space-time dimensions we argue that = 0 for the previously obtained solution of Q chi = 0, Q^{dagger} Psi = 0.
Mirror Symmetry is T-Duality (1996)
Strominger, Andrew, Yau, Shing-Tung, Zaslow, Eric
It is argued that every Calabi-Yau manifold $X$ with a mirror $Y$ admits a family of supersymmetric toroidal 3-cycles. Moreover the moduli space of such cycles together with their flat connections is...
Multiresolution computation of conformal structures of surfaces (1996)
Xianfeng Gu, Yalin Wang, Shing-tung Yau
The conformal structures are only determined by
Mirror Symmetry is T-Duality (1996)
Andrew Strominger, Shing-tung Yau, Eric Zaslow
It is argued that every Calabi-Yau manifold X with a mirror Y admits a family of supersymmetric toroidal 3-cycles. Moreover the moduli space of such cycles together with their flat connections is...
Multiresolution Computation of Conformal Structures of Surfaces (1996)
Xianfeng Gu Yalin, Xianfeng Gu, Yalin Wang, Shing-tung Yau
An e#cient multiresolution method to compute global conformal structures of nonzero genus triangle meshes is introduced. The homology, cohomology groups of meshes are computed explicitly, then a...
BPS States, String Duality, and Nodal Curves on K3 (1995)
We describe the counting of BPS states of Type II strings on K3 by relating the supersymmetric cycles of genus $g$ to the number of rational curves with $g$ double points on K3. The generating...
Mirror Maps, Modular Relations and Hypergeometric Series I (1995)
Lian, Bong H., Yau, Shing-Tung
Motivated by the recent work of Kachru-Vafa in string theory, we study in Part A of this paper, certain identities involving modular forms, hypergeometric series, and more generally series solutions...
Mirror Maps, Modular Relations and Hypergeometric Series II (1995)
Lian, Bong H., Yau, Shing-Tung
As a continuation of \lianyaufour, we study modular properties of the periods, the mirror maps and Yukawa couplings for multi-moduli Calabi-Yau varieties. In Part A of this paper, motivated by the...
Arithmetic Properties of Mirror Map and Quantum Coupling (1994)
Lian, Bong H., Yau, Shing-Tung
We study some arithmetic properties of the mirror maps and the quantum Yukawa coupling for some 1-parameter deformations of Calabi-Yau manifolds. First we use the Schwarzian differential equation,...
Mirror Symmetry, Mirror Map and Applications to Complete Intersection Calabi-Yau Spaces (1994)
Hosono, S., Klemm, A., Theisen, S., Yau, Shing-Tung
We extend the discussion of mirror symmetry, Picard-Fuchs equations, instanton-corrected Yukawa couplings, and the topological one-loop partition function to the case of complete intersections with...
Existence of black hole solutions for the Einstein-Yang/Mills equations (1993)
Wasserman, Arthur G., Yau, Shing-Tung, Smoller, Joel A.
This paper provides a rigorous proof of the existence of an infinite number of black hole solutions to the Einstein-Yang/Mills equations with gauge group SU (2), for any event horizon. It is also...
Smooth static solutions of the Einstein-Yang/Mills equation (1992)
Smoller, Joel, Wasserman, Arthur G., Yau, Shing-Tung, McLeod, J. Bryce
We consider the Einstein/Yang-Mills equations in $3+1$ space time dimensions with $\SU(2)$ gauge group and prove rigorously the existence of a globally defined smooth static solution. We show that...
Smooth static solutions of the Einstein/Yang-Mills equations (1991)
Yau, Shing-Tung, Smoller, Joel A., Wasserman, Arthur G., McLeod, J. B.
We consider the Einstein/Yang-Mills equations in 3+1 space time dimensions with SU (2) gauge group and prove rigorously the existence of a globally defined smooth static solution. We show that the...
An estimate of the gap of the first two eigenvalues in the Schrödinger operator (1985)
Singer, I. M., Wong, Bun, Yau, Shing-Tung, Yau, Stephen S.-T.
Schoen, Richard M., Yau, Shing-Tung
We find some integrability conditions for low-dimensional manifolds to admit metrics with nonnegative scalar curvature. In particular, we solve the positive action conjecture in general relativity in...
Schoen, Richard, Yau, Shing-Tung
We study three-dimensional Riemannian manifolds with nonnegative scalar curvature. We find new topological obstruction for such manifolds. Our method turns out to be useful in studying the positive...
On the structure of complete simply-connected Kähler manifolds with nonpositive curvature
Siu, Yum-Tong, Yau, Shing-Tung
We prove that a complete simply-connected Kähler manifold with nonpositive sectional curvature is biholomorphic to the complex Euclidean space if the curvature is suitably small at infinity.
Calabi's conjecture and some new results in algebraic geometry
We announce a proof of Calabi's conjectures on the Ricci curvature of a compact Kähler manifold and then apply it to prove some new results in algebraic geometry and differential geometry. For...
Schoen, Richard M., Yau, Shing-Tung
We find some integrability conditions for low-dimensional manifolds to admit metrics with nonnegative scalar curvature. In particular, we solve the positive action conjecture in general relativity in...
Schoen, Richard, Yau, Shing-Tung
We study three-dimensional Riemannian manifolds with nonnegative scalar curvature. We find new topological obstruction for such manifolds. Our method turns out to be useful in studying the positive...
On the structure of complete simply-connected Kähler manifolds with nonpositive curvature
Siu, Yum-Tong, Yau, Shing-Tung
We prove that a complete simply-connected Kähler manifold with nonpositive sectional curvature is biholomorphic to the complex Euclidean space if the curvature is suitably small at infinity.
Calabi's conjecture and some new results in algebraic geometry
We announce a proof of Calabi's conjectures on the Ricci curvature of a compact Kähler manifold and then apply it to prove some new results in algebraic geometry and differential geometry. For...
A geometric approach to problems in birational geometry
A classical set of birational invariants of a variety are its spaces of pluricanonical forms and some of their canonically defined subspaces. Each of these vector spaces admits a typical metric...