Calculus of Variations and Geometric Measure Theory
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A. Mondino

The Willmore functional and other $L^p$ curvature functionals in Riemannian manifolds

created by mondino on 18 Oct 2013

[BibTeX]

Phd Thesis

Inserted: 18 oct 2013
Last Updated: 18 oct 2013

Year: 2011

Abstract:

Using techniques both of non linear analysis and geometric measure theory, we prove existence of minimizers and more generally of critical points for the Willmore functional and other $L^p$ curvature functionals for immersions in Riemannian manifolds. More precisely, given a $3$-dimensional Riemannian manifold $(M,g)$ and an immersion of a sphere $f:S^2 \hookrightarrow (M,g)$ we study the following problems. \\

1) The Conformal Willmore functional in a perturbative setting: consider $(M,g)=(R^3,\delta+\epsilon h)$ the euclidean $3$-space endowed with a perturbed metric ($h=h_{\mu\nu}$ is a smooth field of symmetric bilinear forms); we prove, under assumptions on the trace free Ricci tensor and asymptotic flatness, existence of critical points for the Conformal Willmore functional $I(f):=\frac{1}{2} \int
\mathring{A}
^2$ (where $\mathring{A}:=A-\frac{1}{2}H$ is the trace free second fundamental form). The functional is conformally invariant in curved spaces. We also establish a non existence result in general Riemannian manifolds. The technique is perturbative and relies on a Lyapunov-Schmidt reduction. \\

2) The Willmore functional in a semi-perturbative setting: consider $(M,g)=(R^3, \delta+h)$ where $h=h_{\mu\nu}$ is a $C^{\infty}_0(R^3)$ field of symmetric bilinear forms with compact support and small $C^1$ norm. Under a general assumption on the scalar curvature we prove existence of a smooth immersion of $S^2$ minimizing the Willmore functional $W(f):=\frac{1}{4} \int
H
^2$ (where $H$ is the mean curvature). The technique is more global and relies on the direct method in the calculus of variations. \\

3) The functionals $E:=\frac{1}{2} \int
A
^2 $ and $W_1:=\int\left( \frac{
H
^2}{4}+1 \right)$ in compact ambient manifolds: consider $(M,g)$ a $3$-dimensional compact Riemannian manifold. We prove, under global conditions on the curvature of $(M,g)$, existence and regularity of an immersion of a sphere minimizing the functionals $E$ or $W_1$. The technique is global, uses geometric measure theory and regularity theory for higher order PDEs. \\

4) The functionals $E_1:=\int \left( \frac{
A
^2}{2} +1 \right) $ and $W_1:=\int\left( \frac{
H
^2}{4}+1 \right)$ in noncompact ambient manifolds: consider $(M,g)$ a $3$-dimensional asymptotically euclidean non compact Riemannian $3$-manifold. We prove, under general conditions on the curvature of $(M,g)$, existence and regularity of an immersion of a sphere minimizing the functionals $E_1$ or $W_1$. The technique relies on the direct method in the calculus of variations. \\

5) The supercritical functionals $\int
H
^p$ and $\int
A
^p$ in arbitrary dimension and codimension: consider $(N,g)$ a compact $n$-dimensional Riemannian manifold possibly with boundary. For any $2\leq m<n$ consider the functionals $\int
H
^p$ and $\int
A
^p$ with $p>m$, defined on the $m$-dimensional submanifolds of $N$. We prove, under assumptions on $(N,g)$, existence and partial regularity of a minimizer of such functionals in the framework of varifold theory. During the arguments we prove some new monotonicity formulas and new Isoperimetric Inequalities which are interesting by themselves.


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