# Milano - 14th December 2018

## SPEAKERS

ICTP

### Lorenzo Mazzieri

Università di Trento

### Stefano Pigola

Università dell'Insubria

## SCHEDULE

 10:30 ⬤ • 11:00 ⬤ • • • 12:00 ⬤ • • • • • 13:30 ⬤ • • • 14:30 ⬤ • 15:00 ⬤ • • • 16:00 ⬤

## DIRECTIONS

The workshop will take place at:

University of Milano - Bicocca

Department of Mathematics and its Applications

Via R. Cozzi, 55 - Milano

Building U5 - 3rd floor - Room 3014

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#### Coffee break

Participation is free, but we encourage you to register in order to ease the organization of the coffee breaks.

#### Intersection properties of weighted minimal hypersurfaces in the Gaussian space

A celebrated theorem by D. Hoffman and W. Meeks in the 90’s states that two properly immersed minimal surfaces in the Euclidean 3-space must intersect unless they are parallel planes. The result, which fails in higher dimensions, relies on the following half-space property of independent interest: a properly immersed minimal surface contained in a half-space must be planar. If we consider compact minimal hypersurfaces in the sphere or, more generally, in a compact ambient manifold with positive Ricci curvature, the second alternative in Hoffman-Meeks theorem, as well as the constraint on the dimension, disappear. This is a classical result of T. Frankel in the 60’s. In recent years, a lot of attention has been put on weighted minimal submanifolds of the Gaussian space. This ambient space, that shares the non-compactness of the Euclidean space, has a positive constant Bakry-Émery Ricci curvature. A natural question is whether and to what extent Frankel’s result extends to this setting. In this talk I will survey some partial results that include generalized half-space properties in the properly immersed case, and the intersection of properly embedded hypersurfaces sufficiently separated at infinity. It is a joint work with Debora Impera and Michele Rimoldi.

#### Minkowski inequality for mean convex domains

In response to a question raised by Huisken, we prove that the Minkowski Inequality $$\textstyle \big|\partial \Omega\big|^\frac{n-2}{n-1} \, | {\mathbb{S}^{n-1}}|^\frac{1}{n-1} \,\, \leq \,\, \int\limits_{\partial \Omega} \!\frac{\rm H}{n-1} \,\, {\rm d}\sigma \,$$ holds true under the mere assumption that $\Omega$ is a bounded domain with smooth mean convex boundary sitting inside $\mathbb{R}^n$, $n \geq 3$. The result is new even for surfaces in Euclidean three-space, and can be used in this setting to deduce the celebrated De Lellis-Müller nearly umbilical estimates, with a better constant. Our proof relies on a careful analysis of the level set flow of the $p$-capacitary potentials of $\Omega$, as $p \to 1$. (Joint works with V. Agostiniani, M. Fogagnolo and A. Pinamonti).

In response to a question raised by Huisken, we prove that the Minkowski Inequality $$\big|\partial \Omega\big|^{(n-2)/(n-1)} \, | {\mathbb{S}^{n-1}}|^{1/(n-1)} \,\, \leq \,\, \int\limits_{\partial \Omega} \!\frac{\rm H}{n-1} \,\, {\rm d}\sigma \,$$ holds true under the mere assumption that $\Omega$ is a bounded domain with smooth mean convex boundary sitting inside $\mathbb{R}^n$, $n \geq 3$. The result is new even for surfaces in Euclidean three-space, and can be used in this setting to deduce the celebrated De Lellis-Müller nearly umbilical estimates, with a better constant. Our proof relies on a careful analysis of the level set flow of the $p$-capacitary potentials of $\Omega$, as $p \to 1$. (Joint works with V. Agostiniani, M. Fogagnolo and A. Pinamonti).

#### Kaehler constant scalar curvature metrics on blow ups and resolutions of singularities

After recalling the gluing construction for Kaehler constant scalar curvature and extremal (à la Calabi) metrics starting from a compact or ALE orbifolds with isolated singularities, I will show how to compute the Futaki invariant of the adiabatic classes in this setting, extending previous work by Stoppa and Odaka. Besides giving new existence and non-existence results, the connection with the Tian-Yau-Donaldson Conjecture and the K-stability of the resolved manifold will be discussed. The original part of this talk will cover joint works with A. Della Vedova, R. Lena and L. Mazzieri.

REGISTER

Participation is free, but we encourage you to register in order to ease the organization of the coffee breaks. Please send a message to the organizers by email or by filling this form*.

*Pressing 'Send' the content of the 'Message' textbox will be submitted to the organizers. A more structured form, e.g. with name, affiliation and email fields, is not compatible with current privacy rules.

ORGANIZERS

Alberto Della Vedova
alberto.dellavedova@unimib.it

Roberto Paoletti
roberto.paoletti@unimib.it

Università di Milano - Bicocca
Dipartimento di Matematica e Applicazioni
Via R. Cozzi, 55 - 20125 Milano