Related papers: Weighted $\infty$-Willmore Spheres

We study the spectral implications of re-weighting a graph by the $\ell_\infty$-Lewis weights of its edges. Our main motivation is the ER-Minimization problem (Saberi et al., SIAM'08): Given an undirected graph $G$, the goal is to find…

In this note, we extend diameter bounds of Simon, Topping, and Wu--Zheng to submanifolds with boundary and (potentially non-compact) ambient manifolds with minor curvature restrictions. The bound is dependent on both an integral of mean…

In this paper, we recast the variational formulation corresponding to the single layer boundary integral operator $\operatorname{V}$ for the wave equation as a minimization problem in $L^2(\Sigma)$, where $\Sigma := \partial \Omega \times…

We prove that the support of an $ m $ dimensional rectifiable varifold with a uniform lower bound on the density and bounded generalized mean curvature can be covered $ \mathscr{H}^{m} $ almost everywhere by a countable union of $m$…

We consider limits of weakly converging $W^{1,2}$-maps $\Phi_k$ from a ball $B \subset \mathbb{R}^2$ into $\mathbb{R}^3$ which are conformal immersions. Under the assumption that a normal curvature term is small, namely if for the normal…

The aim of this paper is to investigate the differential geometry of immersed surfaces in three-dimensional normed spaces from the viewpoint of affine differential geometry. We endow the surface with a useful Riemannian metric which is…

For a surface immersed in a three-dimensional space endowed with a norm instead of an inner product, one can define analogous concepts of curvature and metric. With these concepts in mind, various questions immediately appear. The aim of…

We consider properly immersed finite topology minimal surfaces S in complete finite volume hyperbolic 3-manifolds N, and in M x S(1), where M is a complete hyperbolic surface of finite area. We prove S has finite total curvature equal to…

In this work we present new fundamental tools for studying the variations of the Willmore functional of immersed surfaces into $R^m$. This approach gives for instance a new proof of the existence of a Willmore minimizing embedding of an…

For any 3-manifold M and any nonnegative integer g, we give here examples of metrics on M each of which has a sequence of embedded minimal surfaces of genus g and without Morse index bounds. On any spherical space form S^3/Gamma we…

We explore the relation among volume, curvature and properness of a $m$-dimensional isometric immersion in a Riemannian manifold. We show that, when the $L^p$-norm of the mean curvature vector is bounded for some $m \leq p\leq \infty$, and…

For an oriented isometric immersion $f:M\to S^n$ the spherical Gauss map is the Legendrian immersion of its unit normal bundle $UM^\perp$ into the unit sphere subbundle of $TS^n$, and the geodesic Gauss map $\gamma$ projects this into the…

Let $P$ be a submanifold properly immersed in a rotationally symmetric manifold having a pole and endowed with a weight $e^h$. The aim of this paper is twofold. First, by assuming certain control on the $h$-mean curvature of $P$, we…

This article investigates stationary surfaces with boundaries, which arise as the critical points of functionals dependent on curvature. Precisely, a generalized "bending energy" functional $\mathcal{W}$ is considered which involves a…

It is shown that gravity can be incorporated into the Standard Model (SM) in a way solving the hierarchy problem. For this, the SM effective action in flat spacetime is adapted to curved spacetime via not only the general covariance but…

The isometric immersion of two-dimensional Riemannian manifolds or surfaces with negative Gauss curvature into the three-dimensional Euclidean space is studied in this paper. The global weak solutions to the Gauss-Codazzi equations with…

We discuss infinitesimal isometries of the middle surfaces and present some characteristic conditions for a function to be the normal component of an infinitesimal isometry. Our results show that those characteristic conditions depend on…

Let $n \ge 2$, $p \in (1, \, +\infty)$ be given and let $\Sigma$ be a $n$-dimensional, closed hypersurface in $\mathbb{R}^{n+1}$. Denote by $A$ its second fundamental form, and by $\mathring{A}$ the tensor $A - \frac{1}{n} A^i_i g$ where $g…

We establish upper and lower universal bounds for potentials of weighted designs on the sphere $\mathbb{S}^{n-1}$ that depend only on quadrature nodes and weights derived from the design structure. Our bounds hold for a large class of…

Let $M$ be a compact connected oriented $n-1$ dimensional manifold without boundary. In this work, shape space is the orbifold of unparametrized immersions from $M$ to $\mathbb R^n$. The results of \cite{Michor118}, where mean curvature…