[CvGmt News] seminario Cabré e Carlotto 11/06

Daniele Castorina castorin at mat.uniroma2.it
Wed Jun 5 10:34:28 CEST 2013

Cari amici,

questa settimana avremo DUE seminari di ED a Tor Vergata..attenzione 
all'orario e mi raccomando la puntualità:

Martedi' 11 Giugno 2013, h 14:15, Aula D’ Antoni
Alessandro Carlotto - Stanford University & MSRI
"Scalar curvature rigidity phenomena in asymptotically flat spaces." In 
this talk, I will outline my recent proof of the following statement:
An asymptotically flat 3-manifold (M,g) of nonnegative scalar curvature 
that contains a complete (non compact), properly embedded stable minimal 
surface is isometric to the Euclidean space. Since the nonnegativity of 
the scalar curvature is implied by the (time-symmetric) Einstein 
constraint equations, the previous result naturally applies to initial 
data sets in General Relativity.
The proof of this theorem is based on a characterization of finite index 
minimal surfaces, on classical infinitesimal rigidity results by 
Fischer-Colbrie and Schoen and on the positive mass theorem by 
Schoen-Yau. A key technical step is the improvement of the decay rate of 
the second fundamental form of such surface: this follows from a 
tilt-excess decay lemma at infinity which exploits ideas of 
Allard-Almgren and Schoen-Simon. The talk will be aimed at the general 
mathematical audience, with plenty of references to other recent results 
and open problems in the field.

Martedi' 11 Giugno 2013, h 15:15, Aula D' Antoni
Xavier Cabré (ICREA and UPC, Barcelona)
"Sharp isoperimetric inequalities via the ABP method" We prove some old 
and new isoperimetric inequalities with the best constant via the ABP 
method. More precisely, we obtain a new family of sharp isoperimetric 
inequalities with weights (or densities) in open convex cones of 
$\mathbb{R}^n$. Our results apply to all nonnegative homogeneous weights 
satisfying a concavity condition in the cone. Surprisingly, even that 
our weights are not radially symmetric, Euclidean balls centered at the 
origin (intersected with the cone) minimize the weighted isoperimetric 
quotient. As a particular case of our results, we provide with new 
proofs of classical results such as the Wulff inequality and the 
isoperimetric inequality in convex cones of Lions and Pacella. 
Furthermore, we also study the anisotropic isoperimetric problem for the 
same class of weights and we prove that the Wulff shape always minimizes 
the anisotropic weighted perimeter under the weighted volume constraint.

Daniele Castorina
Dipartimento di Matematica - Studio 1221
Università di Roma "Tor Vergata"
Via della Ricerca Scientifica 00133 Roma
email: castorin at mat.uniroma2.it
tel: +390672594653

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