# Abstracts for Young Women in Geometry

Alternatively have a look at the program.

## Registration

Posted in
Date:
Mon, 2017-04-03 09:30 - 10:00
Location:
MPIM Lecture Hall
Parent event:
Young Women in Geometry

## Free boundary minimal surfaces and the Steklov eigenvalue problem (I)

Posted in
Speaker:
Ailana M. Fraser
Affiliation:
University of British Columbia
Date:
Mon, 2017-04-03 10:00 - 11:00
Location:
MPIM Lecture Hall
Parent event:
Young Women in Geometry

If we fix a smooth compact surface $M$ with boundary, we can consider
all Riemannian metrics on $M$ with fixed boundary length. We can then
hope to find a canonical metric by maximizing a first eigenvalue. The
eigenvalue problem which turns out to lead to geometrically interesting
maximizing metrics is the Steklov eigenvalue problem; that is, the
Dirichlet-to-Neumann map on $\partial M$. There is a close connection
between this eigenvalue problem and minimal surfaces in a Euclidean ball
that are proper in the ball and meet the boundary of the ball

## Evolution Equations in Geometry (I)

Posted in
Speaker:
Esther Cabezas-Rivas
Affiliation:
Universität Frankfurt
Date:
Mon, 2017-04-03 11:30 - 12:30
Location:
MPIM Lecture Hall
Parent event:
Young Women in Geometry

Evolution equations have been used to address successfully key questions in Differential Geometry
like isoperimetric inequalities, the Poincaré conjecture, Thurston’s geometrization conjecture, or
the differentiable sphere theorem.

During this series of lectures we will give a general introduction to geometric flows, which are sort of
non-linear versions of the heat equation for a relevant geometric quantity. These equations should be
understood as a tool to canonically deform a manifold into a manifold with nicer properties, for instance,

## Geometric spectral invariants (I)

Posted in
Speaker:
Julie Rowlett
Affiliation:
Chalmers University of Technology
Date:
Mon, 2017-04-03 14:30 - 15:30
Location:
MPIM Lecture Hall
Parent event:
Young Women in Geometry

We will begin with the discovery of the first geometric spectral invariants by Hermann Weyl in 1912.
In the first lecture, we will recall Weyl's original proof using Dirichlet-Neumann bracketing, to demonstrate
"das asymptotische verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen,'' which is
now known as Weyl's law.  In the second lecture, we will turn to the spectral invariant which lead to the
discovery of new geometric spectral invariants in the 1960s and 1980s:  the heat trace.  We will recall the

## A spectral lower bound in singular setting

Posted in
Speaker:
Ilaria Mondello
Affiliation:
Université Paris Est Créteil
Date:
Mon, 2017-04-03 15:30 - 16:00
Location:
MPIM Lecture Hall
Parent event:
Young Women in Geometry

A very well known result in Riemannian geometry, the Obata-Lichnerowicz theorem, relates the Ricci curvature and the spectrum of the Laplacian: for a compact Riemannian manifold of dimension n, if the Ricci tensor is bounded below by $(n-1)$, then the first non-zero eigenvalue of the Laplacian is larger than or equal to the dimension of the manifold. Equality holds if and only if the manifold is isometric to the sphere.

## Four-dimensional Cohomogeneity one Ricci flow and Nonnegative sectional curvature

Posted in
Speaker:
Anusha Mangala Krishnan
Affiliation:
University of Pennsylvania
Date:
Mon, 2017-04-03 17:00 - 17:30
Location:
MPIM Lecture Hall
Parent event:
Young Women in Geometry

We study the Ricci flow on $4$-manifolds admitting a cohomogeneity one group action, i.e. an isometric group action such that the orbit space $M/G$ is $1$-dimensional. We use this to demonstrate the first examples of 4-manifolds having nonnegative sectional curvature which under the Ricci flow, immediately acquire some negatively curved 2-planes. In particular, we show that $S^4$, $\mathbb{C}P^2$, $S^2\times S^2$ and $\mathbb{C}P^2 \# \overline{\mathbb{C}P^2}$ admit such metrics. This talk is based on joint work with Renato G. Bettiol.

## Selberg zeta function for hyperbolic orbifolds: transfer approach

Posted in
Speaker:
Ksenia Fedosova
Organiser(s):
Chalmers University of Technology
Date:
Mon, 2017-04-03 17:30 - 18:00
Location:
MPIM Lecture Hall
Parent event:
Young Women in Geometry

To a hyperbolic surface and a finite-dimensional representation of its fundamental group, we associate a Selberg zeta function. The main goal of the talk is to show that under certain conditions, the Selberg zeta function admits a meromorphic extension to the whole complex plane. Our main tool is the use of transfer operators. This is a joint work with Anke Pohl.

## Evolution Equations in Geometry (II)

Posted in
Speaker:
Esther Cabezas-Rivas
Organiser(s):
Universität Frankfurt
Date:
Tue, 2017-04-04 09:00 - 10:00
Location:
MPIM Lecture Hall
Parent event:
Young Women in Geometry

Evolution equations have been used to address successfully key questions in Differential Geometry
like isoperimetric inequalities, the Poincaré conjecture, Thurston’s geometrization conjecture, or
the differentiable sphere theorem.

During this series of lectures we will give a general introduction to geometric flows, which are sort of
non-linear versions of the heat equation for a relevant geometric quantity. These equations should be
understood as a tool to canonically deform a manifold into a manifold with nicer properties, for instance,

## The Willmore energy of non-orientable surfaces

Posted in
Speaker:
Elena Mäder-Baumdicker
Affiliation:
KIT Karlsruhe
Date:
Tue, 2017-04-04 10:00 - 10:30
Location:
MPIM Lecture Hall
Parent event:
Young Women in Geometry

The Willmore conjecture was proved by F.\ Marques and A.\ Neves in 2012. Since then we
know that the Clifford torus has the lowest Willmore energy among all tori in $\mathbb R^3$.
I will present results concerning immersed Klein bottles in  $\mathbb R^n$ with low Willmore
energy, $n\geq 4$. For Klein bottles immersed in $\mathbb R^4$ it is known that there are three
distinct regular homotopy classes each one containing an embedding. I will explain that
one of these classes contains the embedded Klein bottle with lowest Willmore energy among

## Holonomy Groups of Locally Conformally Kähler Metrics

Posted in
Speaker:
Michaela Pilca
Affiliation:
Universität Regensburg
Date:
Tue, 2017-04-04 11:00 - 11:30
Location:
MPIM Lecture Hall
Parent event:
Young Women in Geometry

A Hermitian metric on a complex manifold is locally conformally Kähler
if it is conformal to a Kähler metric in the neighbourhood of each point.
In this talk will be given the classification of the possible holonomy groups
of such metrics on compact manifolds. As a by-product, we present the geometric
description of conformal classes on compact manifolds containing two non-homothetic
Kähler metrics - necessarily with respect to non-conjugate complex structures.
The talk is based on joint work with Farid Madani and Andrei Moroianu, which is

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