Weekly seminar links: Geometry
The current research of the differential geometry and geometric analysis group explores the areas of: canonical metrics on Kähler manifolds, geometric curvature flows, geometric problems in general relativity, nonlinear Hamiltonian systems and closed geodesics on manifolds, submanifold geometry and Finsler geometry, etc. The group members include Gang Tian, Yuguang Shi, Xiaohuan Mo, Xiaohua Zhu, Xiang Ma, Wei Wang and Bin Zhou. The group has made a large number of remarkable progress in rencent years.
1. Canonical Metrics on Kähler Manifolds (TIAN Gang, ZHU Xiaohua, ZHOU Bin )
The existence of canonical metrics on Kähler manifolds, including Kähler-Einstein metrics, Kähler-Ricci solitons, constant scalar curvature metrics, extremal metrics, etc, is a fundamental problem in Kähler geometry. A famous folklore conjecture in this field, called Yau-Tian-Donaldson conjecture is: A compact Kähler manifold admits canonical metrics if and only if it is stable in sense of geometric invariant theory.
In his talk at SBU in 2012, Prof. Gang Tian announced the resolution of the existence of Kähler-Einstein on Fano manifolds and gave an outlined proof. This confirms the Yau-Tian-Donaldson conjecture in Fano case: A Fano manifold admits Kähler Einstein metrics if and only if it is K-stable. This result has been published in [Tian, Comm. Pure Appl. Math. (2015)].
Meanwhile, in a joint work with Bing Wang, Gang Tian investigated the compactness and regularity theory for the Gromov-Hausdorff limit space of a sequence of almost Einstein manifolds. They proved a deep structure theorem, i.e., the regularity set is a smooth, convex, open manifold and the singular set is codimension 2. This result also plays an important role in the proof of Yau-Tian-Donaldson conjecture. The paper has been published in [Tian-Wang, J. Amer. Math. Soc. (2015)].
Another progress is made in the study of Kähler-Ricci solitons. Feng Wang, Bin Zhou and Xiaohua Zhu extended Tian-Zhu’s modified Futaki invariant to general test configurations of the manifold and gave a new proof of the existence of Kähler-Ricci solitons on toric Fano manifolds [Wang-Zhou-Zhu, Adv. Math. (2016)]. This new invariant can be seen as the generalization of Ding-Tian invariant and Donadson-Futaki invariant to Kähler-Ricci case.
2. Geometric Curvature Flows (TIAN Gang, ZHU Xiaohua)
Ricci flow was introduced by R. Hamilton in 1982. In 2003, Perelman solved the well-known Poincaré conjecture by handling singularities of Ricci flow in dimension 3. Now Ricci flow has become one of the central subjects in modern geometry. Gang Tian made important contribution to the verification of Pereman’s proof of Poincaré conjecture and Geometrization conjecture. Together with John Morgan, Tian finished the monograph《Ricci Flow and Poincaré Conjecture》. In this book, they gave a detailed exposition of Pereman’s work and also elaborated some conception of their own. Geometric curvature flows, especially Kähler-Ricci flow is one of their main research subjects.
1) Kähler-Ricci Flow and Hamilton-Tian Conjecture
In Kähler case, the famous Hamilton-Tian conjecture asserts that Kähler-Ricci flow on a Fano manifold will converges to a Kähler-Ricci soliton(probably with singular set of low dimension). In 2007, Gang Tian and Xiaohua Zhu proved the convergence of normalized Kähler-Ricci flow on any Kähler-Einstein manifold without holomorphic vector fields [Tian-Zhu, J. Amer. Math. Soc. (2007)]. Then in 2013, they removed the assumption on holomorphic vector fields and confirmed the convergence on manifolds with Kähler-Ricci solitons [Tian-Zhu, J. Reine Angew. Math. (2013); Tian-Zhang- Zhang-Zhu, Trans. Amer. Math. Soc. (2013)]. Recently, Gang Tian and Zhenlei Zhang have settled the convergence of Kähler-Ricci flow on general Fano manifolds in dimension 2 and 3. As a consequence, they gave a new proof of Yau-Tian-Donaldson in dimension 2 and 3. This paper has been published in [Tian-Zhang, Acta Math. (2016)].
2) Non-Kähler Flows
In non-Kähler geometry, directly running Ricci flow does not keep the corresponding geometric structure. For this reason, generalizations of Kähler-Ricci flow arise in recent years. J. Streets and Gang Tian have introduced a series of non-Kähler geometric flows since 2008, including pluriclosed flow, Hermitian curvature flow, almost Hermitian curvature flow, symplectic curvature flow, pluriclosed flow on generalized Kähler manifolds [Streets-Tian, J. Eur. Math. Soc. (JEMS) (2011); Streets- Tian, J. Reine Angew. Math. (2014); Streets-Tian, Nuclear Physics B (2012)]. These works open up a new research field.
3) Analytic Minimal Model Problem
With the development of Kähler-Ricci flow, Gang Tian has proposed a program on the study of Analytic Minimal Model Program. In a joint work with Jian Song, he proved that on any algebraic manifold, there exists a canonical measure which is birationally invariant. In fact, the measure can be derived through Kähler-Ricci flow [Song-Tian, J. Amer. Math. Soc. (2012)]. This solves a conjecture by Tsuji. Very recently, they proved the existence of weak Kähler-Ricci flow on any affine variety with log terminal singularities. Furthermore the flow can be uniquely continued through divisorial contractions and flips if it exists [Song-Tian, Invent. Math. (2017)]. Based on these results, it is expected that the study on Minimal Model Program and birational geometry by Kähler-Ricci flow will be a new popular subject.
3. Geometric Problems in General Relativity (SHI Yuguang )
General relativity is one of the main motivations of differential geometry. Yuguang Shi’s research focuses on the study of geometric problems related with quasi-local mass in general relativity. He and his collaborators obtained a generalization of Brown-York mass and proved its positivity [Shi-Tam, Class. Quantum Grav. (2007)]; they investigated the limit of various quasi-local mass, and proved that on an asymptotically flat (AF) manifold, the Brown-York mass and Hawking mass of the coordinates spheres tends to the ADM mass of this AF manifold when the domains enclosed by these coordinates spheres approach to the whole manifold [Fan-Shi-Tam, Comm. Anal. Geom. (2009)]. Shi also obtained an area comparison theorem of isoperimetric surfaces in 3-dimensional AF manifolds with nonnegative scalar curvature. One application of this result is to show the positivity of isoperimetric mass introduced by Husiken [Shi, Int. Math. Res. Not. IMRN (2016)].
Conformally compact Einstein (CCE) manifolds are basic objects in AdS/CFT corresponding theory. In a joint work with Gang Li and Jie Qing, Yuguang Shi studied some geometric properties of CCE manifolds. Among other things, they proved that when the Yamabe quotient of the conformal infinity boundary of this CCE manifold is close enough to that of the standard sphere, then the sectional curvature of the CCE manifold tends to −1. As a corollary, the underlying manifold is diffeomorphic to the Euclidean ball. As an application, they obtained that all CCE manifolds with the standard sphere as their conformal infinity boundaries are isometric to the hyperbolic space [Li-Qing-Shi, Trans. Amer. Math. Soc. (2017)].
4. Nonlinear Hamiltonian Systems and Closed Geodesics on Manifolds (WANG Wei)
1) Nonlinear Hamiltonian Systems
In the study of of nonlinear Hamiltonian system, Wei Wang focus on the problem of multiplicity and stability of closed characteristics on compact convex hypersurfaces inR2n. A longstanding conjecture in this area based on the work of A. Weinstein says the number of closed characteristics on compact convex hypersurfaces inR2n is no less than n. In [Wang-Hu-Long, Duke Math. J. (2007)], Wei Wang and his collaborators solved this conjecture when n=3. One of the main ingredients of the proof is a new resonance identity for closed characteristics. Recently, Wei Wang solved the conjecture when n=4 and obtain a new lower bound for the number of closed characteristicsn+12+1 [Wang, Adv. Math. (2016)]. With the new methods in these works, he established many results on the stability of closed characteristics.
2) Closed Geodesics on Manifolds
The problem of closed geodesics on compact manifolds causes many interests. It is well-known that the origin of Morse theory is the study of geodesics. Wei Wang mainly considers the problem of closed geodesics on compact Riemannian or Finsler manifolds. A famous conjecture on the number of closed geodesics on compact Riemannian manifolds is: There exist infinitely many closed geodesics on every compact Riemannian manifold. This conjecture has been proved except for several cases which include the compact rank one symmetric spaces, e.g., M=Sn, CPn, HPn, CaP2. In contrast with the Riemannian case, the conjecture becomes false when one moves to the Finsler case due to the counterexamples by A. B. Katok in 1973. Based on this, D. V. Anosov proposed the following conjecture in the 1974’s ICM: There exist at least 2n+12 closed geodesics on every Finsler n-sphere(Sn, F), i.e., the lower bound for the number of closed geodesics is given by that of Katok’s examples. Wei Wang proved that there exist at least 2n+12 closed geodesics on (Sn, F) for any bumpy Finsler n-sphere (Sn, F) with reversibilityλ, if its flag curvature K satisfies(λλ+1)2<K≤1[Wang, Adv. Math. (2012)]. The result has been extended to general Finsler manifolds with bumpy property [Duan-Long-Wang, J. Differential Geom. (2016)], which gives a confirmed answer to a generalized version of Anosov’s conjecture for a generic case. Some stability results have also been obtained [Wang, Math. Ann. (2013); Wang, J. Differential Geom. (2015)].
5. Submanifold Geometry (MA Xiang)
Submanifold geomtry is a traditional research field of geometry. Recently, Xiang Ma obtained important results on Willmore surfaces and other submanifolds in Lorentz space forms and Möbius geometry.
1) Willmore Surfaces
Xiang Ma (jointly with Changping Wang and Peng Wang) generalized the classification results of Bryant, Ejiri, Montiel etc. on Willmore 2-spheres [Ma-Wang- Wang, J. Differential Geom. (2017)]. Before this breakthrough, 20 years have passed without significant progress on this difficult problem. They gave a classification theorem, showing that the so-called adjoint transform of Willmore surfaces really works on this problem. On the other hand, they constructed concrete examples, showing that a Willmore 2-sphere in higher codimensional space does not necessarily have a dual surface. This ongoing work will hopefully give a complete classification for any codimensions.
2) Lorentz Geometry
In the study of Lorentz geometry, Prof. Xiang Ma and his collaborators generalized the theory of minimal surfaces in 3-dimensional Euclidean space and that of maximal surfaces in 3-dimensional Lorentz space, unifying them in the 4-dimensional Lorentz space. They provided Weierstrass-type representation formulas and Gauss-Bonnet type formula of the total curvature. In particular, they discovered new phenomena on embeddedness property and on the singular ends [Ma-Wang- Wang, Adv. Math. (2013)]. Recently, they generalized convex hypersurfaces (ovaloids) to higher codimensional spacelike submanifolds in Lorentz spaces. In the 1-dimensional case, for the so-called strong spacelike closed curves with winding number 1, they obtained the reversed Fenchel-type inequality. Alongside this work, they generalized the classical Crofton formula and the solutions of the Plateau problem [Ye-Ma-Wang, Ann. Global Anal. Geom. (2016)].
3) Möbius Geometry
In Möbius geometry, Xiang Ma and Tongzhu Li, Changping Wang continued the work of Cartan and Dajczer on the rigidity/deformation phenomenon of isometric immersed hypersurfaces; they discussed analogous problems on Möbius metric and obtained complete and similar results [Li-Ma-Wang, Adv. Math. (2014)]. Another interesting topic is Wintgen ideal submanifolds. These objects attain equality in a universal inequality (the so-called DDVV inequality), thus are certain kind of extremal submanifolds. This property is also conformal invariant, which allows them to study them in Möbius geometry. By introduceing the conformal Gauss map, they discovered that in codimension-2 case, the image of this map degenerates to a holomorphic curve in the target manifold (a quadric hypersurface in the complex projective space); conversely, one can start from such holomorphic curves and construct all such codimension-2 Wintgen ideal submanifolds [Li-Ma-Wang-Xie, Tohoku Math. J. (2016)]. More generally, they obtained a series of classification theorems, enriching and deepening the knowledge of the structure of these submanifolds.
6. Finsler Geometry (MO Xiaohuan)
Finsler geometry is a direction highly recommended by Profesor S.S. Chen. Around Finsler geometry, Xiaohuan Mo obtained a sequence of important results in the study of flag curvature and Weyl curvature in recent years.
The flag curvature is the most important Riemannian geometry quantity on Finsler manifolds, which is the generalization of the sectional curvature on Riemannian manifolds. Xiaohuan Mo and his collaborators identified the changing formula of the flag curvature for the navigation problem. They first set up a non-increasing formula of the flag curvature for a similar navigation problem. Based on this, they determined the flag curvature of the new Finsler metric obtained by any Finsler metric and any conformal field through the navigation problem, and constructed a new non-trivial example [Huang-Mo, Pacific J. Math. (2015)]. The results include those of Dawei Bao, Robles, Zhongmin Shen, Xinyue Cheng, Foulon et al. The construction and classification of Finsler metrics are two basic problems in Finsler geometry, especially for Finsler manifolds with constant flag curvature. Prof. Xiaoxuan Mo and his collaborators completely classified the projectively flat, spherical symmetric Finsler metrics with constant flag curvature [Mo-Zhu, Internat. J. Math. (2012)]. He has proved that the space of the constant flag curvature Finsler surface with two-dimensional equidistant groups depends on two arbitrary constants. Very recently, Xiaohuan Mo and R. Bryant prove that the normal form of the 2-dimensional Finsler metric of the regular flag curvature with a killing field depends on two univariable functions [Bryant-Liu-Mo, J. Geom. Phys. (2017)].
Weyl curvature is an important quantity on Finsler manifolds in sense of projective geometry. Metrics with the quadratic Weyl curvature are of interest when studying Weyl curvature. Xiaohuan Mo and his collaborators succeeded in finding a large number of examples of Finsler metrics with non-trivial quadratic Weyl curvature. These include Randers metrics with non-constant flag curvature and Finsler metrics with quadratic Weyl Curvature and non-quadratic Riemannian curvature [Liu-Mo, Math. Nachr. (2015)].