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

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 $f:\Sp^2 \hookrightarrow (M,g)$ we study the following problems. 1) The Conformal Willmore functional in a perturbative setting: consider $(M,g)=(\Rtre,\eu+\epsilon h)$ 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 |A^\circ|^2 $ (where $A^\circ:=A-\frac{1}{2}H$ 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 $(M,g)=(\Rtre, \eu+h)$ where $h=h_{\mu\nu}$ is a $C^{\infty}_0(\Rtre)$ 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 $\Sp^2$ 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.