We consider the initial-boundary value problem
of the normalized Ricci flow on a given compact
2-D surfaces with boundary, with prescribed geodesic
curvature on the boundary. We establish a large class
of initial-boundary data, so that the flow exists for
all $t>0$, and converges locally uniformly to the
complete hyperbolic metric in the interior of the
surface.
Title: Wulff inequality for minimal submanifolds in Euclidean space
Abstract: Geometric inequality for minimal submanifolds in Euclidean space plays
very important role in geometric analysis. In this talk, we will talk about recent work,
that we prove a Wulff inequality for $n$-dimensional minimal submanifolds with boundary in $R^{n+m}$,
where we associate a nonnegative anisotropic weight to the boundary of minimal submanifolds.
The Wulff inequality constant depends only on $m$ and $n$, and is independent of the weights.
The inequality is sharp if $m = 1, 2$ and the weight is the support function of ellipsoids or
certain type of centrally symmetric long convex bodies. This is a joint work with Wenkui Du and Yuchao Yi.
Title: Interior control for surfaces with positive scalar curvature and its application
Abstract: Let $M^{n}$,
$n\in\{3,4,5\}$, be a closed aspherical
$n$-manifold and $S\subset M$ a subset
consisting of disjoint incompressible
embedded closed aspherical submanifolds
(possibly with different dimensions).
When $n =3,4$, we show that $M\setminus S$
cannot admit any complete metric with
positive scalar curvature.
The key ingredient is a new interior
control for the extrinsic diameter of
surfaces with positive scalar curvature.
This is joint work with Jianchun Chu
(Peking University) and Jintian Zhu
(Westlake University).
Speaker: Dr. Shihang He (Peking University)
Title:
Positive mass theorems on singular spaces and some applications
Abstract: We establish positive mass theorems on singular spaces under
the assumption that the scalar curvature is nonnegative in the strong spectral sense.
As an application, we obtain desingularization results for strongly stable minimal
hypersurfaces in asymptotically flat (AF) 8-manifolds. These results reveal
certain positive effect of the minimal hypersurface singularities in scalar
curvature geometry, which also leads to an alternative proof of the positive
mass theorem in dimension 8. This is joint work with Professors Yuguang Shi and Haobin Yu.
Title: On the topology of manifolds with positive intermediate curvature
Abstract: In this talk, I will introduce a conjecture relating
the topology of a manifold's universal cover with the existence of metrics
with positive m-intermediate curvature. We prove the result in dimension 3~5
and for most choices of m in dimension 6. As a corollary, we show that a
closed aspherical 6-manifold cannot admit a metric with positive 4-intermediate
curvature. This talk is base on the joint work with Liam Mazurowski and Xuan Yao.
Title: Rigidity Theorem for Poincare-Einstein Manifolds
Abstract: In this talk, I first introduce the classical rigidity
theorem for Poincare-Einstein manifold, which has conformal compactification in
high regularity. Then I will report some recent rigidity result for Poincare-Einstein
manifold in the upper half plane model, which take the Euclidean space as
its conformal infinity and whose adapted conformal metric has quadratic curvature
decay at infinity. This is joint work with Sanghoon Lee (KIAS).
Speaker: Dr. Yipeng Wang (Columbia University)
Title: Rigidity Results Involving Stabilized Scalar Curvature
Abstract: Gromov introduced the notion of stabilized scalar
curvature, which arises naturally in the context of warped product extensions.
This concept also appears in the study of the geometry of weighted manifolds
and in Perelman's work on the Ricci flow. In this talk, I will explore the
relationship between various formulations of stabilized scalar curvature
and explain how several classical scalar curvature rigidity results can be
extended to this more general setting.
Speaker: Philipp Reiser (University of Fribourg)
Title: Manifolds of positive Bakry-Emery Ricci curvature
Abstract: The Bakry-Emery Ricci tensor is a generalization of
the classical Ricci tensor to the setting of weighted Riemannian manifolds,
i.e. Riemannian manifolds whose Riemannian volume forms are weighted by a
smooth function. In analogy with important open problems in the Riemannian
case, we consider the question of which manifolds admit a weighted Riemannian
metric of positive Bakry-Emery Ricci curvature. We will show that surgery
techniques that are used in the Riemannian case can be extended and improved
in the weighted case. As application we will construct weighted Riemannian
metrics of positive Bakry-Emery Ricci curvature on all closed,
simply-connected spin 5-manifolds. This is joint work with Francesca Tripaldi.
Title: $3$-Manifolds with positive scalar curvature and bounded geometry
Abstract: In this talk we will work towards proving the following
theorem: a contractible $3$-manifold with positive scalar curvature and bounded
geometry must be diffeomorphic to $R^3$. The proof involves running an innermost
weak inverse mean curvature flow on the manifold. This talk is based on joint
work with Otis Chodosh and Yi Lai.
Speaker: Junrong Yan (Northeastern University)
Title: Heat Kernel Expansion and Weyl's
Law for Schrodinger-Type Operators on Noncompact Manifolds
Abstract: Motivated by the study of Landau-Ginzburg models
in string theory from the viewpoint of index theorem,
we explore the heat kernel expansion for Schrodinger-type operators
on noncompact manifolds. This expansion leads to a local index theorem
for such operators.
Unlike in the compact case, the heat kernel in the noncompact
setting exhibits new behaviors. Obtaining its precise expansion and deriving a
remainder estimate require careful analysis. We will first present our
approach to establishing this expansion.
As a key application, we study Weyl's law for such operators.
In the compact case, such results follow from Karamata's Tauberian
theorem, but the standard Tauberian argument does not apply in the
noncompact setting. To address this, we develop a new version of Karamata's theorem.
This is joint work with Xianzhe Dai.
Title: Rigidity of the Positive Mass Theorems
Abstract: Schoen-Yau and Witten demonstrated that
the total mass $m$ of asymptotically flat initial data sets(IDS)
in general relativity are non-negative. Combining spinorial methods
with spacetime harmonic functions, we prove that if $m=0$, the IDS
must embed into a pp-wave spacetime. This resolves the rigidity
conjecture for the spacetime positive mass theorem on spin manifolds
in all dimensions. Furthermore, adapting these techniques to
the hyperboloidal setting, we establish that asymptotically
hyperboloidal IDS with zero mass must embed isometrically
into Minkowski space, contrasting with the asymptotically flat case.
It is joint work with Sven Hirsch and Hyun Chul Jang.
16. 2025 Jun 4, 9-10:30 AM,
Tencent Meeting