[Dottorcomp] Fwd: corso di dottorato "Elliptic PDEs and hypersurface geometry"

[CLAUDIA CONTARDI]

---------- Forwarded message ---------
Da: Stefano Pigola <stefano.pigola a unimib.it>
Date: lun 2 set 2024 alle ore 07:40
Subject: corso di dottorato "Elliptic PDEs and hypersurface geometry"
To: <g.barbieri43 a campus.unimib.it>, <m.fusari a campus.unimib.it>, Simone
Gallivanone <s.gallivanone a campus.unimib.it>, <e.marmo a campus.unimib.it>, <
f.mastrogiacomo2 a campus.unimib.it>, Tommaso Toti <t.toti a campus.unimib.it>,
<giulia.cavalleri01 a universitadipavia.it>, <
claudia.contardi01 a universitadipavia.it>, <
paolo.grossi01 a universitadipavia.it>, <antonio.lacopo01 a universitadipavia.it>,
<edoardogiovann.tolotti01 a universitadipavia.it>, <
luca.vai01 a universitadipavia.it>, <mario.cajumi01 a universitadipavia.it>, <
gabriele.castellari01 a universitadipavia.it>, <
lorenzo.fassina02 a universitadipavia.it>, <
massimiliano.ghiotto01 a universitadipavia.it>, <
chiara.perinati01 a universitadipavia.it>, <
mariateresa.ruggiero01 a universitadipavia.it>, Paolo Malanchini <
p.malanchini a campus.unimib.it>, <c.nocita a campus.unimib.it>, <
e.sforza a campus.unimib.it>, <k.haile a campus.unimib.it>, Anna Donadini <
a.donadini a campus.unimib.it>, <j.feuerpfeil a campus.unimib.it>


Cari/e tutti/e,

scrivo per sapere se vi siano interessati al corso “Elliptic PDEs and
hypersurface geometry” che dovrebbe tenersi a partire da metà ottobre o giù
di lì. La durata del corso è di 28h. Non ho gli indirizzi dei nuovi
immatricolati: nel caso li conosciate vi sarei grato se poteste girar loro
il messaggio.

Il corso è inteso come introduzione ad alcune tecniche dell’analisi
geometrica in spazi lisci. Ho pensato ad un programma di massima (forse
sovradeterminato). È riportato qui sotto.

Chi fosse interessato per favore me lo faccia sapere rispondendo a questo
messaggio. Potremmo anche organizzare un incontro per discutere più in
dettaglio il programma e i tempi.

Grazie.
A presto,

Stefano

***

1. *Introduction: the origin of a technique*. From L.V. Ahlfors to S.T. Yau

2. *Hypersurfaces in Euclidean space. *Basic concepts and formulas.

3. *Height estimates for compact H-graphs*. A-priori estimates of the
height of a Constant Mean Curvature (CMC) compact graph with planar
boundary. The 2-d case is a celebrated result by J. Serrin in 1969.
Following W. Meeks we shall give a proof in n-dimensions using the maximum
principle for the Laplace-Beltrami operator.

4. *Minimality of entire H-graphs and the Bernstein problem*. Entire
H-graphs are minimal, as one can see via “isoperimetric” considerations.
This is a toy case of an open problem by M.P. do Carmo in the 80s. Entire
minimal graphs are hyperplanes (Bernstein problem) in dimensions strictly
less than 8. We shall discuss a 2-dimensional proof using the notion of
“parabolicity” in potential theory.

5. *The half-space property of entire minimal graphs*. In any dimension,
the only entire minimal graphs contained  in a vertical half-space are the
hyperplanes. This is a celebrated result by E. Bombieri, E. De Giorgi and
M. Miranda in 1969. We shall present the “elementary" proof by Nick
Korevaar (1986) via gradient estimates.

If time permits:

6. *Bounded minimal surfaces. *A great impulse in minimal surface theory
was given by the E. Calabi conjectures stating that some / all the
coordinate functions of a complete minimal surface in Euclidean 3-space
must be unbounded. In general the conjectures are false (L. Jorge and F.
Xavier in 1980 and N. Nadirashvili in 1996). However, they are true under
topological assumptions (as a consequence of the recent theory by T.
Colding and W. Mincozzi) or volume/curvature restrictions. We shall discuss
boundedness properties of (the “ends”  of) a minimal surface by means of
extrinsic radius estimates for CMC surfaces. These are obtained using
maximum principles at infinity and decay of positive bounded solutions of
semilinear PDEs.




Università di Milano-Bicocca
Dipartimento di Matematica e Applicazioni
via Roberto Cozzi, 55
20125 Milano