Related papers: Rigidity and non-rigidity results for conformal im…

We study immersed tori in $3$-space minimizing the Willmore energy in their respective conformal class. Within the rectangular conformal classes $\;(0,b)\;$ with $\;b \sim 1\;$ the homogenous tori $\;f^b\;$ are known to be the unique…

We consider surfaces in ${\mathbb R}^3$ of type ${\mathbb S}^2$ which minimize the Willmore functional with prescribed isoperimetric ratio. The existence of smooth minimizers was proved by Schygulla (Archive Rational Mechanics and Analysis,…

Let (M,g) be a four or six dimensional compact Riemannian manifold which is locally conformally flat and assume that its boundary is totally umbilical. In this note, we prove that if the Euler characteristic of M is equal to 1 and if its…

This paper is devoted to rigidity results for some elliptic PDEs and related interpolation inequalities of Sobolev type on smooth compact connected Riemannian manifolds without boundaries. Rigidity means that the PDE has no other solution…

We prove a rigidity theorem in Poisson geometry around compact Poisson submanifolds, using the Nash-Moser fast convergence method. In the case of one-point submanifolds (fixed points), this immediately implies a stronger version of Conn's…

We introduce a notion of generalized Willmore functionals motivated by the Hawking energy of General Relativity and bending energies of membranes. An example of a bending energy is discussed in detail. Using results of Y. Chen and J. Li, we…

We propose an improved effective-medium theory to obtain the concentration dependence of the viscosity of particle suspensions at arbitrary volume fractions. Our methodology can be applied, in principle, to any particle shape as long as the…

Motivated by a simple model for elastic cell membranes, we minimize the Willmore functional among two-dimensional spheres embedded in R^3 with prescribed isoperimetric ratio.

In this paper we show rigidity results for super-solutions to fully nonlinear elliptic conformally invariant equations on subdomains of the standard $n$-sphere $\mathbb S^n$ under suitable conditions along the boundary. We emphasize that…

Quaternionic analysis, which describes conformal maps from Riemann surfaces into $\mathbb{R}^3$ or $\mathbb{R}^4$, is extended to weakly conformal maps. As a consequence we present a new proof that on any compact Riemann surface $X$ the…

We consider local CR-immersions of a strictly pseudoconvex real hypersurface $M\subset\bC^{n+1}$, near a point $p\in M$, into the unit sphere $\mathbb S\subset\bC^{n+d+1}$ with $d>0$. Our main result is that if there is such an immersion…

We give a survey of various rigidity results involving scalar curvature. Many of these results are inspired by the positive mass theorem in general relativity. In particular, we discuss the recent solution of Min-Oo's Conjecture for the…

We consider isometric immersions of complete connected Riemannian manifolds into space forms of nonzero constant curvature. We prove that if such an immersion is compact and has semi-definite second fundamental form, then it is an embedding…

We study the umbilic points of Willmore surfaces in codimension 1 from the viewpoint of the conformal Gauss map. We first study the local behaviour of the conformal Gauss map near umbilic curves and prove that they are geodesics up to a…

We argue that the standard classification of isometric deformations into infinitesimal v.s. finite is inadequate for the study of compliant shell mechanisms. Indeed, many compliant shells, particularly ones that are periodically corrugated,…

We show certain rigidity for minimizers of generalized Colding-Minicozzi entropies. The proofs are elementary and work even in situations where the generalized entropies are not monotone along mean curvature flow.

We study ends of an oriented, immersed, non-compact, complete Willmore surfaces, which are critical points of the integral of the square of the mean curvature, in asymptotically flat spaces of any dimension; assuming the surface has…

We consider the $L^2$ gradient flow for the Willmore functional in Riemannian manifolds of bounded geometry. In the euclidean case E.\;Kuwert and R.\;Sch\"atzle [\textsl{Gradient flow for the Willmore functional,} Comm. Anal. Geom., 10:…

We give examples of proper minimal immersions in Euclidean space with very rapid area growth. The first is a proper embedding into $\bf{R}^4$ that yields a stable minimal surface, while the second is a proper immersion into $\bf{R}^3$.…

We derive an infinitesimal rigidity lemma for the strain tensor of surfaces with their curvatures changing sign. As an application, we obtain the optimal constant in the first Korn inequality scales like $h^{4/3}$ for such shells of mixed…