Let $M$ be an open (i.e. complete and noncompact)
manifold with nonnegative Ricci curvature. We prove that, under the
conic at infinity condition, if the infimum of the volume growth
order of $M$ equals $k$, then there exists an asymptotic cone of
$M$ whose upper box dimension is at most $k$. We also extend or
partially extend Sormani's results concerning $M$ with linear
volume growth to more relaxed volume growth conditions.
Title: Stability of Ricci flow and
Volume Comparison for Finite-volume Hyperbolic $3$-manifolds
Abstract: In this talk, I will discuss a joint work
with Franco Vargas Pallete on finite-volume hyperbolic $3$-manifolds.
If the initial metric is close to the hyperbolic metric $g_0$, then
the normalized Ricci-DeTurck flow exists for all time and converges
exponentially fast to $g_0$ in a weighted Holder norm.
Furthermore, I will also present an application to volume comparison,
which shows that among metrics with scalar curvature bounded
below by $-6$, $g_0$ minimizes the volume.
Speaker: Dr. Anushree Das (Peking University)
Title:
Volume growth of manifolds under curvature constraints
Abstract:
This talk concerns the question of whether the topology of an
open manifold restricts the volume growth of the metrics
it can admit, specially with nice curvature conditions. For
an open manifold $M$ with one end, and a function $v$ with
bounded growth of derivative, there exists a Riemannian metric
of bounded geometry on $M$ such that the volume growth
function lies in the same growth class as $v$, as proved by
R. Grimaldi and P. Pansu. We prove this in the case of manifolds
with multiple ends and call the constructed metrics
Grimaldi-Pansu metrics. We give uniform explicit bounds
for the volume growth function of these metrics in terms of the
given function in the case of a certain class of manifolds, and
investigate some geometric properties of the Grimaldi-Pansu
metrics to better understand them. Next, we consider the question
in the positive scalar curvature setting. For a function $v$ with
bounded growth of derivative, whether $M$ that admits some metric
of positive scalar curvature also admits a metric of positive
scalar curvature with volume growth of the same growth type as $v$
is unknown. We answer this question positively in the case of
those manifolds which are infinite connected sums of closed manifolds
that admit metrics of positive scalar curvature, and generalise
it to some other settings.
Title: On the Geometry and Uniqueness of Asymptotically Locally Hyperbolic Static Vacuum Black Holes
Abstract:
The classical Minkowski inequality (1903) gives a sharp lower bound
for the total mean curvature of a closed convex surface in Euclidean
space. Using inverse mean curvature flow (IMCF), Guan-Li (2007)
showed that the Minkowski inequality holds for star-shaped
mean-convex surfaces. In 2012, Brendle-Hung -Wang established
the Minkowski inequality on anti-de Sitter Schwarzschild space.
The key in their proof is a new monotonicity formula based on
the Heintze-Karcher inequality.
I will discuss a joint work with Brian Harvie in which we
generalize the Minkowski inequality to 3-dimensional
asymptotically locally hyperbolic static manifolds and use
it to obtain some black hole uniqueness theorems.
Speaker: Dr. Xuwen Zhang (Freiburg University)
Title: Anisotropic minimal graphs with free boundary
Abstract: Minimal surface equation is a classical topic
in Geometric Analysis and PDEs. In this talk, we discuss recent
progress on anisotropic minimal surface equation, and prove the
following Liouville-type theorem: any anisotropic minimal graph with
free boundary in the half-space must be flat, provided that the graph
function has at most one-side linear growth. The linear growth
assumption is sharp, in view of the recent examples constructed by
Connor Mooney-Yang Yang (The anisotropic Bernstein problem, Invent.
Math. 235 (2024), no. 1, 211--232).
This is a joint work with Guofang Wang, Wei Wei, and Chao Xia.