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.

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)].