Huai-Dong Cao - Academia.edu (original) (raw)

Papers by Huai-Dong Cao

arXiv (Cornell University), Jul 28, 2016

We study the fast diffusion equation with a linear forcing term, ∂u ∂t = div(|u| p-1 ∇u) + Ru, (0... more We study the fast diffusion equation with a linear forcing term, ∂u ∂t = div(|u| p-1 ∇u) + Ru, (0.1) under the Ricci flow on complete manifold M with bounded curvature and nonnegative curvature operator, where 0 < p < 1 and R = R(x, t) is the evolving scalar curvature of M at time t. We prove Aronson-Bénilan and Li-Yau-Hamilton type differential Harnack estimates for positive solutions of (0.1). In addition, we use similar method to prove certain Li-Yau-Hamilton estimates for the heat equation and conjugate heat equation which extend those obtained by X. Cao and R. Hamilton [10], X. Cao [9], and S. Kuang and Q. Zhang [23] to noncompact setting.

arXiv (Cornell University), Apr 3, 2023

In this paper, we continue investigating the second variation of Perelman's ν-entropy for compact... more In this paper, we continue investigating the second variation of Perelman's ν-entropy for compact shrinking Ricci solitons. In particular, we improve some of our previous work in [12], as well as the more recent work in [34], and obtain a necessary and sufficient condition for a compact shrinking Ricci soliton to be linearly stable. Our work also extends similar results of Hamilton, Ilmanen and the first author in [9] (see also [10]) for positive Einstein manifolds to the compact shrinking Ricci soliton case.

arXiv (Cornell University), Apr 29, 1999

One of the methods to obtain Frobenius manifold structures is via DGBV (differential Gerstenhaber... more One of the methods to obtain Frobenius manifold structures is via DGBV (differential Gerstenhaber-Batalin-Vilkovisky) algebra construction. An important problem is how to identify Frobenius manifold structures constructed from two different DGBV algebras. For DGBV algebras with suitable conditions, we show the functorial property of a construction of deformations of the multiplicative structures of their cohomology. In particular, we show that quasi-isomorphic DGBV algebras yield identifiable Frobenius manifold structures.

arXiv (Cornell University), Jun 23, 2020

In this article, we study four-dimensional complete gradient shrinking Ricci solitons. We prove t... more In this article, we study four-dimensional complete gradient shrinking Ricci solitons. We prove that a four-dimensional complete gradient shrinking Ricci soliton satisfying a pointwise condition involving either the self-dual or anti-self-dual part of the Weyl tensor is either Einstein, or a finite quotient of either the Gaussian shrinking soliton R 4 , or S 3 × R, or S 2 × R 2. In addition, we provide some curvature estimates for four-dimensional complete gradient Ricci solitons assuming that its scalar curvature is suitable bounded by the potential function.

arXiv (Cornell University), Mar 23, 2009

In this paper we derive a precise estimate on the growth of potential functions of complete nonco... more In this paper we derive a precise estimate on the growth of potential functions of complete noncompact shrinking solitons. Based on this, we prove that a complete noncompact gradient shrinking Ricci soliton has at most Euclidean volume growth. The latter result can be viewed as an analog of the well-known theorem of Bishop that a complete noncompact Riemannian manifold with nonnegative Ricci curvature has at most Euclidean volume growth.

arXiv (Cornell University), Jan 15, 2015

In this paper we consider the Martin compactification, associated with the operator L = ∆ − 1, of... more In this paper we consider the Martin compactification, associated with the operator L = ∆ − 1, of a complete non-compact surface Σ 2 with negative curvature. In particular, we investigate positive eigenfunctions with eigenvalue one of the Laplace operator ∆ of Σ 2 and prove a uniqueness result: such eigenfunctions are unique up to a positive constant multiple if they vanish on the part of the geometric boundary S∞(Σ 2) of Σ 2 where the curvature is bounded above by a negative constant, and satisfy some growth estimate on the other part of S∞(Σ 2) where the curvature approaches zero. This uniqueness result plays an essential role in our recent paper [CH] in which we prove an infinitesimal rigidity theorem for deformations of certain three-dimensional collapsed gradient steady Ricci soliton with a non-trivial Killing vector field. Contents 1. Introduction 1 2. Preliminaries 6 2.1. Martin compactification of complete Riemannian manifolds 6 2.2. Positive L-harmonic functions and Green's function 7 3. Geometric compactification of the surface Σ 2 8 4. The minimal Green's function of the operator L = ∆ − 1 12 5. Asymptotic expansions of the Green's function 22 5.1. Asymptotic expansion at η = 0 23 5.2. Asymptotic expansion at η = ∞ 25 5.3. Asymptotic expansions along the rays η = m |ξ| 30 6. The Martin kernel and Martin boundary 33 7. Proof of Theorem 1.8 37 Appendix A. Singular Sturm-Liouville problem and a Spectral Theorem 39 Appendix B. A second order linear ordinary differential equation 43 References 45

Metric Geometry and Harmonic Functions

One of the most eminent of contemporary mathematicians, Shing-Tung Yau has received numerous hono... more One of the most eminent of contemporary mathematicians, Shing-Tung Yau has received numerous honors, including the 1982 Fields Medal, considered the highest honor in mathematics, for his work in differential geometry. He is known also for his work in algebraic and Kahler geometry, general relativity, and string theory. His influence in the development and establishment of these areas of research has been great. These five volumes reproduce a comprehensive selection of his published mathematical papers of the years 1971 to 1991-a period of groundbreaking accomplishments in numerous disciplines including geometric analysis, Kahler geometry, and general relativity. The editors have organized the contents of this collection by subject area-metric geometry and minimal submanifolds; metric geometry and harmonic functions; eigenvalues and general relativity; and Kahler geometry. Also presented are expert commentaries on the subject matter, and personal reminiscences that shed light on the development of the ideas which appear in these papers.

Metric Geometry and Minimal Submanifolds

One of the most eminent of contemporary mathematicians, Shing-Tung Yau has received numerous hono... more One of the most eminent of contemporary mathematicians, Shing-Tung Yau has received numerous honors, including the 1982 Fields Medal, considered the highest honor in mathematics, for his work in differential geometry. He is known also for his work in algebraic and Kahler geometry, general relativity, and string theory. His influence in the development and establishment of these areas of research has been great. These five volumes reproduce a comprehensive selection of his published mathematical papers of the years 1971 to 1991-a period of groundbreaking accomplishments in numerous disciplines including geometric analysis, Kahler geometry, and general relativity. The editors have organized the contents of this collection by subject area-metric geometry and minimal submanifolds; metric geometry and harmonic functions; eigenvalues and general relativity; and Kahler geometry. Also presented are expert commentaries on the subject matter, and personal reminiscences that shed light on the development of the ideas which appear in these papers.

arXiv (Cornell University), Feb 28, 2014

In this paper we study the porous medium equation (PME) coupled with the Ricci flow on complete m... more In this paper we study the porous medium equation (PME) coupled with the Ricci flow on complete manifolds with bounded nonnegative curvature. In particular, we derive Aronson-Bénilan and Li-Yau-Hamilton type differential Harnack estimates for positive solutions to the PME, with a linear forcing term, under the Ricci flow.

arXiv (Cornell University), Jan 3, 2011

In this paper, we prove a classification theorem for self-shrinkers of the mean curvature flow wi... more In this paper, we prove a classification theorem for self-shrinkers of the mean curvature flow with |A| 2 ≤ 1 in arbitrary codimension. In particular, this implies a gap theorem for self-shrinkers in arbitrary codimension.

arXiv (Cornell University), Dec 8, 2014

The only known example of collapsed three-dimensional complete gradient steady Ricci solitons so ... more The only known example of collapsed three-dimensional complete gradient steady Ricci solitons so far is the 3D cigar soliton N 2 ×R, the product of Hamilton's cigar soliton N 2 and the real line R with the product metric. R. Hamilton has conjectured that there should exist a family of collapsed positively curved three-dimensional complete gradient steady solitons, with S 1-symmetry, connecting the 3D cigar soliton. In this paper, we make the first initial progress and prove that the infinitesimal deformation at the 3D cigar soliton is non-essential. In Appendix A, we show that the 3D cigar soliton is the unique complete nonflat gradient steady Ricci soliton in dimension three that admits two commuting Killing vector fields.

Matemática Contemporânea

In this survey paper, we analyse and compare the recent curvature estimates for three types of 4-... more In this survey paper, we analyse and compare the recent curvature estimates for three types of 4-dimensional gradient Ricci solitons, especially between Ricci shrinkers [58] and expanders [17]. In addition, we provide some new curvature estimates for 4dimensional gradient steady Ricci solitons, including the sharp curvature estimate |Rm| ≤ CR for gradient steady Ricci solitons with positive Ricci curvature (see Theorem 5.2).

In this paper we prove that any n-dimensional (n ≥ 4) complete Bach-flat gradient steady Ricci so... more In this paper we prove that any n-dimensional (n ≥ 4) complete Bach-flat gradient steady Ricci soliton with positive Ricci curvature is isometric to the Bryant soliton. We also show that a three-dimensional gradient steady Ricci soliton with divergence-free Bach tensor is either flat or isometric to the Bryant soliton. In particular, these results improve the corresponding classification theorems for complete locally conformally flat gradient steady Ricci solitons in [6, 9].

arXiv (Cornell University), Apr 9, 2013

Following [CHI], in this paper we study the linear stability of Perelman's νentropy on Einstein m... more Following [CHI], in this paper we study the linear stability of Perelman's νentropy on Einstein manifolds with positive Ricci curvature. We observe the equivalence between the linear stability restricted to the transversal traceless symmetric 2-tensors and the stability of Einstein manifolds with respect to the Hilbert action. As a main application, we give a full classification of linear stability of the ν-entropy on symmetric spaces of compact type. In particular, we exhibit many more linearly stable and linearly unstable examples than previously known and also the first linearly stable examples, other than the standard spheres, whose second variations are negative definite.

Geometry and topology : lectures given at the Geometry and Topology Conferences at Harvard University in 2011 and at Lehigh University in 2012

This volume includes papers presented by several speakers at the Geometry and Topology conference... more This volume includes papers presented by several speakers at the Geometry and Topology conferences at Harvard University in 2011 and at Lehigh University in 2012. Included are works by Simon Brendle, on the Lagrangian minimal surface equation and related problems; by Sergio Cecotti and Cumrun Vafa, concerning classification of complete N=2 supersymmetric theories in four dimensions; by F. Reese Harvey and H. Blaine Lawson Jr., on existence, uniqueness, and removable singularities for non-linear PDEs in geometry; by Janos Kollar, concerning links of complex analytic singularities; by Claude LeBrun, on Calabi energies of extremal toric surfaces; by Mu-Tao Wang, concerning mean curvature flows and isotopy problems; and by Steve Zelditch, on eigenfunctions and nodal sets.

Eigenvalues and general relativity

One of the most eminent of contemporary mathematicians, Shing-Tung Yau has received numerous hono... more One of the most eminent of contemporary mathematicians, Shing-Tung Yau has received numerous honors, including the 1982 Fields Medal, considered the highest honor in mathematics, for his work in differential geometry. He is known also for his work in algebraic and Kahler geometry, general relativity, and string theory. His influence in the development and establishment of these areas of research has been great. These five volumes reproduce a comprehensive selection of his published mathematical papers of the years 1971 to 1991-a period of groundbreaking accomplishments in numerous disciplines including geometric analysis, Kahler geometry, and general relativity. The editors have organized the contents of this collection by subject area-metric geometry and minimal submanifolds; metric geometry and harmonic functions; eigenvalues and general relativity; and Kahler geometry. Also presented are expert commentaries on the subject matter, and personal reminiscences that shed light on the ...

Calculus of Variations and Partial Differential Equations

Let (M n , g, f), n ≥ 5, be a complete gradient expanding Ricci soliton with nonnegative Ricci cu... more Let (M n , g, f), n ≥ 5, be a complete gradient expanding Ricci soliton with nonnegative Ricci curvature Rc ≥ 0. In this paper, we show that if the asymptotic scalar curvature ratio of (M n , g, f) is finite (i.e., lim sup r→∞ Rr 2 < ∞), then the Riemann curvature tensor must have at least sub-quadratic decay, namely, lim sup r→∞ |Rm| r α < ∞ for any 0 < α < 2. † Research partially supported by a Simons Foundation Collaboration Grant (#586694 HC).

In this paper, we derive curvature estimates for 4-dimensional complete gradient expanding Ricci ... more In this paper, we derive curvature estimates for 4-dimensional complete gradient expanding Ricci solitons with nonnegative Ricci curvature (outside a compact set K). More precisely, we prove that the norm of the curvature tensor Rm and its covariant derivative ∇Rm can be bounded by the scalar curvature R by |Rm| ≤ CaRa and |∇Rm| ≤ CaRa (on M \K), for any 0 ≤ a < 1 and some constant Ca > 0. Moreover, if the scalar curvature has at most polynomial decay at infinity, then |Rm| ≤ CR (on M \K). As an application, it follows that that if a 4-dimensional complete gradient expanding Ricci soliton (M, g, f) has nonnegative Ricci curvature and finite asymptotic scalar curvature ratio then it has finite asymptotic curvature ratio, and C asymptotic cones at infinity (0 < α < 1) according to Chen-Deruelle [20].

The Journal of Geometric Analysis

In this paper we define a Weil-Petersson type metric on the space of shrinking Kähler-Ricci solit... more In this paper we define a Weil-Petersson type metric on the space of shrinking Kähler-Ricci solitons and prove a necessary and sufficient condition on when it is independent of the choices of Kähler-Ricci soliton metrics. We also show that the Weil-Petersson metric is Kähler when it defines a metric on the Kuranishi space of small deformations of Kähler-Ricci solitons. Finally, we establish the first and second order deformation of Fano Kähler-Ricci solitons and show that, essentially, the first effective term in deforming Kähler-Ricci solitons leads to the Weil-Petersson metric.

Asian Journal of Mathematics, 2006

In this paper, we give a complete proof of the Poincaré and the geometrization conjectures. This ... more In this paper, we give a complete proof of the Poincaré and the geometrization conjectures. This work depends on the accumulative works of many geometric analysts in the past thirty years. This proof should be considered as the crowning achievement of the Hamilton-Perelman theory of Ricci flow.

arXiv (Cornell University), Jul 28, 2016

We study the fast diffusion equation with a linear forcing term, ∂u ∂t = div(|u| p-1 ∇u) + Ru, (0... more We study the fast diffusion equation with a linear forcing term, ∂u ∂t = div(|u| p-1 ∇u) + Ru, (0.1) under the Ricci flow on complete manifold M with bounded curvature and nonnegative curvature operator, where 0 < p < 1 and R = R(x, t) is the evolving scalar curvature of M at time t. We prove Aronson-Bénilan and Li-Yau-Hamilton type differential Harnack estimates for positive solutions of (0.1). In addition, we use similar method to prove certain Li-Yau-Hamilton estimates for the heat equation and conjugate heat equation which extend those obtained by X. Cao and R. Hamilton [10], X. Cao [9], and S. Kuang and Q. Zhang [23] to noncompact setting.

arXiv (Cornell University), Apr 3, 2023

In this paper, we continue investigating the second variation of Perelman's ν-entropy for compact... more In this paper, we continue investigating the second variation of Perelman's ν-entropy for compact shrinking Ricci solitons. In particular, we improve some of our previous work in [12], as well as the more recent work in [34], and obtain a necessary and sufficient condition for a compact shrinking Ricci soliton to be linearly stable. Our work also extends similar results of Hamilton, Ilmanen and the first author in [9] (see also [10]) for positive Einstein manifolds to the compact shrinking Ricci soliton case.

arXiv (Cornell University), Apr 29, 1999

One of the methods to obtain Frobenius manifold structures is via DGBV (differential Gerstenhaber... more One of the methods to obtain Frobenius manifold structures is via DGBV (differential Gerstenhaber-Batalin-Vilkovisky) algebra construction. An important problem is how to identify Frobenius manifold structures constructed from two different DGBV algebras. For DGBV algebras with suitable conditions, we show the functorial property of a construction of deformations of the multiplicative structures of their cohomology. In particular, we show that quasi-isomorphic DGBV algebras yield identifiable Frobenius manifold structures.

arXiv (Cornell University), Jun 23, 2020

In this article, we study four-dimensional complete gradient shrinking Ricci solitons. We prove t... more In this article, we study four-dimensional complete gradient shrinking Ricci solitons. We prove that a four-dimensional complete gradient shrinking Ricci soliton satisfying a pointwise condition involving either the self-dual or anti-self-dual part of the Weyl tensor is either Einstein, or a finite quotient of either the Gaussian shrinking soliton R 4 , or S 3 × R, or S 2 × R 2. In addition, we provide some curvature estimates for four-dimensional complete gradient Ricci solitons assuming that its scalar curvature is suitable bounded by the potential function.

arXiv (Cornell University), Mar 23, 2009

In this paper we derive a precise estimate on the growth of potential functions of complete nonco... more In this paper we derive a precise estimate on the growth of potential functions of complete noncompact shrinking solitons. Based on this, we prove that a complete noncompact gradient shrinking Ricci soliton has at most Euclidean volume growth. The latter result can be viewed as an analog of the well-known theorem of Bishop that a complete noncompact Riemannian manifold with nonnegative Ricci curvature has at most Euclidean volume growth.

arXiv (Cornell University), Jan 15, 2015

In this paper we consider the Martin compactification, associated with the operator L = ∆ − 1, of... more In this paper we consider the Martin compactification, associated with the operator L = ∆ − 1, of a complete non-compact surface Σ 2 with negative curvature. In particular, we investigate positive eigenfunctions with eigenvalue one of the Laplace operator ∆ of Σ 2 and prove a uniqueness result: such eigenfunctions are unique up to a positive constant multiple if they vanish on the part of the geometric boundary S∞(Σ 2) of Σ 2 where the curvature is bounded above by a negative constant, and satisfy some growth estimate on the other part of S∞(Σ 2) where the curvature approaches zero. This uniqueness result plays an essential role in our recent paper [CH] in which we prove an infinitesimal rigidity theorem for deformations of certain three-dimensional collapsed gradient steady Ricci soliton with a non-trivial Killing vector field. Contents 1. Introduction 1 2. Preliminaries 6 2.1. Martin compactification of complete Riemannian manifolds 6 2.2. Positive L-harmonic functions and Green's function 7 3. Geometric compactification of the surface Σ 2 8 4. The minimal Green's function of the operator L = ∆ − 1 12 5. Asymptotic expansions of the Green's function 22 5.1. Asymptotic expansion at η = 0 23 5.2. Asymptotic expansion at η = ∞ 25 5.3. Asymptotic expansions along the rays η = m |ξ| 30 6. The Martin kernel and Martin boundary 33 7. Proof of Theorem 1.8 37 Appendix A. Singular Sturm-Liouville problem and a Spectral Theorem 39 Appendix B. A second order linear ordinary differential equation 43 References 45

Metric Geometry and Harmonic Functions

One of the most eminent of contemporary mathematicians, Shing-Tung Yau has received numerous hono... more One of the most eminent of contemporary mathematicians, Shing-Tung Yau has received numerous honors, including the 1982 Fields Medal, considered the highest honor in mathematics, for his work in differential geometry. He is known also for his work in algebraic and Kahler geometry, general relativity, and string theory. His influence in the development and establishment of these areas of research has been great. These five volumes reproduce a comprehensive selection of his published mathematical papers of the years 1971 to 1991-a period of groundbreaking accomplishments in numerous disciplines including geometric analysis, Kahler geometry, and general relativity. The editors have organized the contents of this collection by subject area-metric geometry and minimal submanifolds; metric geometry and harmonic functions; eigenvalues and general relativity; and Kahler geometry. Also presented are expert commentaries on the subject matter, and personal reminiscences that shed light on the development of the ideas which appear in these papers.

Metric Geometry and Minimal Submanifolds

One of the most eminent of contemporary mathematicians, Shing-Tung Yau has received numerous hono... more One of the most eminent of contemporary mathematicians, Shing-Tung Yau has received numerous honors, including the 1982 Fields Medal, considered the highest honor in mathematics, for his work in differential geometry. He is known also for his work in algebraic and Kahler geometry, general relativity, and string theory. His influence in the development and establishment of these areas of research has been great. These five volumes reproduce a comprehensive selection of his published mathematical papers of the years 1971 to 1991-a period of groundbreaking accomplishments in numerous disciplines including geometric analysis, Kahler geometry, and general relativity. The editors have organized the contents of this collection by subject area-metric geometry and minimal submanifolds; metric geometry and harmonic functions; eigenvalues and general relativity; and Kahler geometry. Also presented are expert commentaries on the subject matter, and personal reminiscences that shed light on the development of the ideas which appear in these papers.

arXiv (Cornell University), Feb 28, 2014

In this paper we study the porous medium equation (PME) coupled with the Ricci flow on complete m... more In this paper we study the porous medium equation (PME) coupled with the Ricci flow on complete manifolds with bounded nonnegative curvature. In particular, we derive Aronson-Bénilan and Li-Yau-Hamilton type differential Harnack estimates for positive solutions to the PME, with a linear forcing term, under the Ricci flow.

arXiv (Cornell University), Jan 3, 2011

In this paper, we prove a classification theorem for self-shrinkers of the mean curvature flow wi... more In this paper, we prove a classification theorem for self-shrinkers of the mean curvature flow with |A| 2 ≤ 1 in arbitrary codimension. In particular, this implies a gap theorem for self-shrinkers in arbitrary codimension.

arXiv (Cornell University), Dec 8, 2014

The only known example of collapsed three-dimensional complete gradient steady Ricci solitons so ... more The only known example of collapsed three-dimensional complete gradient steady Ricci solitons so far is the 3D cigar soliton N 2 ×R, the product of Hamilton's cigar soliton N 2 and the real line R with the product metric. R. Hamilton has conjectured that there should exist a family of collapsed positively curved three-dimensional complete gradient steady solitons, with S 1-symmetry, connecting the 3D cigar soliton. In this paper, we make the first initial progress and prove that the infinitesimal deformation at the 3D cigar soliton is non-essential. In Appendix A, we show that the 3D cigar soliton is the unique complete nonflat gradient steady Ricci soliton in dimension three that admits two commuting Killing vector fields.

Matemática Contemporânea

In this survey paper, we analyse and compare the recent curvature estimates for three types of 4-... more In this survey paper, we analyse and compare the recent curvature estimates for three types of 4-dimensional gradient Ricci solitons, especially between Ricci shrinkers [58] and expanders [17]. In addition, we provide some new curvature estimates for 4dimensional gradient steady Ricci solitons, including the sharp curvature estimate |Rm| ≤ CR for gradient steady Ricci solitons with positive Ricci curvature (see Theorem 5.2).

In this paper we prove that any n-dimensional (n ≥ 4) complete Bach-flat gradient steady Ricci so... more In this paper we prove that any n-dimensional (n ≥ 4) complete Bach-flat gradient steady Ricci soliton with positive Ricci curvature is isometric to the Bryant soliton. We also show that a three-dimensional gradient steady Ricci soliton with divergence-free Bach tensor is either flat or isometric to the Bryant soliton. In particular, these results improve the corresponding classification theorems for complete locally conformally flat gradient steady Ricci solitons in [6, 9].

arXiv (Cornell University), Apr 9, 2013

Following [CHI], in this paper we study the linear stability of Perelman's νentropy on Einstein m... more Following [CHI], in this paper we study the linear stability of Perelman's νentropy on Einstein manifolds with positive Ricci curvature. We observe the equivalence between the linear stability restricted to the transversal traceless symmetric 2-tensors and the stability of Einstein manifolds with respect to the Hilbert action. As a main application, we give a full classification of linear stability of the ν-entropy on symmetric spaces of compact type. In particular, we exhibit many more linearly stable and linearly unstable examples than previously known and also the first linearly stable examples, other than the standard spheres, whose second variations are negative definite.

Geometry and topology : lectures given at the Geometry and Topology Conferences at Harvard University in 2011 and at Lehigh University in 2012

This volume includes papers presented by several speakers at the Geometry and Topology conference... more This volume includes papers presented by several speakers at the Geometry and Topology conferences at Harvard University in 2011 and at Lehigh University in 2012. Included are works by Simon Brendle, on the Lagrangian minimal surface equation and related problems; by Sergio Cecotti and Cumrun Vafa, concerning classification of complete N=2 supersymmetric theories in four dimensions; by F. Reese Harvey and H. Blaine Lawson Jr., on existence, uniqueness, and removable singularities for non-linear PDEs in geometry; by Janos Kollar, concerning links of complex analytic singularities; by Claude LeBrun, on Calabi energies of extremal toric surfaces; by Mu-Tao Wang, concerning mean curvature flows and isotopy problems; and by Steve Zelditch, on eigenfunctions and nodal sets.

Eigenvalues and general relativity

One of the most eminent of contemporary mathematicians, Shing-Tung Yau has received numerous hono... more One of the most eminent of contemporary mathematicians, Shing-Tung Yau has received numerous honors, including the 1982 Fields Medal, considered the highest honor in mathematics, for his work in differential geometry. He is known also for his work in algebraic and Kahler geometry, general relativity, and string theory. His influence in the development and establishment of these areas of research has been great. These five volumes reproduce a comprehensive selection of his published mathematical papers of the years 1971 to 1991-a period of groundbreaking accomplishments in numerous disciplines including geometric analysis, Kahler geometry, and general relativity. The editors have organized the contents of this collection by subject area-metric geometry and minimal submanifolds; metric geometry and harmonic functions; eigenvalues and general relativity; and Kahler geometry. Also presented are expert commentaries on the subject matter, and personal reminiscences that shed light on the ...

Calculus of Variations and Partial Differential Equations

Let (M n , g, f), n ≥ 5, be a complete gradient expanding Ricci soliton with nonnegative Ricci cu... more Let (M n , g, f), n ≥ 5, be a complete gradient expanding Ricci soliton with nonnegative Ricci curvature Rc ≥ 0. In this paper, we show that if the asymptotic scalar curvature ratio of (M n , g, f) is finite (i.e., lim sup r→∞ Rr 2 < ∞), then the Riemann curvature tensor must have at least sub-quadratic decay, namely, lim sup r→∞ |Rm| r α < ∞ for any 0 < α < 2. † Research partially supported by a Simons Foundation Collaboration Grant (#586694 HC).

In this paper, we derive curvature estimates for 4-dimensional complete gradient expanding Ricci ... more In this paper, we derive curvature estimates for 4-dimensional complete gradient expanding Ricci solitons with nonnegative Ricci curvature (outside a compact set K). More precisely, we prove that the norm of the curvature tensor Rm and its covariant derivative ∇Rm can be bounded by the scalar curvature R by |Rm| ≤ CaRa and |∇Rm| ≤ CaRa (on M \K), for any 0 ≤ a < 1 and some constant Ca > 0. Moreover, if the scalar curvature has at most polynomial decay at infinity, then |Rm| ≤ CR (on M \K). As an application, it follows that that if a 4-dimensional complete gradient expanding Ricci soliton (M, g, f) has nonnegative Ricci curvature and finite asymptotic scalar curvature ratio then it has finite asymptotic curvature ratio, and C asymptotic cones at infinity (0 < α < 1) according to Chen-Deruelle [20].

The Journal of Geometric Analysis

In this paper we define a Weil-Petersson type metric on the space of shrinking Kähler-Ricci solit... more In this paper we define a Weil-Petersson type metric on the space of shrinking Kähler-Ricci solitons and prove a necessary and sufficient condition on when it is independent of the choices of Kähler-Ricci soliton metrics. We also show that the Weil-Petersson metric is Kähler when it defines a metric on the Kuranishi space of small deformations of Kähler-Ricci solitons. Finally, we establish the first and second order deformation of Fano Kähler-Ricci solitons and show that, essentially, the first effective term in deforming Kähler-Ricci solitons leads to the Weil-Petersson metric.

Asian Journal of Mathematics, 2006

In this paper, we give a complete proof of the Poincaré and the geometrization conjectures. This ... more In this paper, we give a complete proof of the Poincaré and the geometrization conjectures. This work depends on the accumulative works of many geometric analysts in the past thirty years. This proof should be considered as the crowning achievement of the Hamilton-Perelman theory of Ricci flow.