Related papers: A classification theorem for Helfrich surfaces

In this article we classify the totally umbilical surfaces which are immersed into a wide class of Riemannian manifolds having a structure of warped product, more precisely, we show that a totally umbilical surface immersed into the warped…

We dominate non-integral singular operators by adapted sparse operators and derive optimal norm estimates in weighted spaces. Our assumptions on the operators are minimal and our result applies to an array of situations, whose prototype are…

We establish the following Hadamard--Stoker type theorem: Let $f:M^n\rightarrow\mathscr{H}^n\times\mathbb R$ be a complete connected hypersurface with positive definite second fundamental form, where $\mathscr H^n$ is a Hadamard manifold.…

We study 2-dimensional submanifolds of the space ${\mathbb{L}}({\mathbb{H}}^3)$ of oriented geodesics of hyperbolic 3-space, endowed with the canonical neutral K\"ahler structure. Such a surface is Lagrangian iff there exists a surface in…

In this paper we analyse the structure of the spaces of smooth type functions, generated by elements of arbitrary Hilbert spaces, as a continuation of the research in our previous papers in this series. We prove that these spaces are…

We give a H\"ormander type $L^2-$estimate for the $\bar{\partial}-$equation with respect to the measure $\delta_\Omega^{-\alpha}dV$, $\alpha<1$, on any bounded pseudoconvex domain with $C^2-$boundary. Several applications to the function…

We classify simply-connected, complete Willmore surfaces with vanishing Gaussian curvature. We also study the Willmore cones and give a classification. As an application, we give a Bernstein-type theorem.

Given a $C^k$-smooth closed embedded manifold $\mathcal N\subset{\mathbb R}^m$, with $k\ge 2$, and a compact connected smooth Riemannian surface $(S,g)$ with $\partial S\neq\emptyset$, we consider $\frac 12$-harmonic maps $u\in…

We give sufficient and necessary geometric conditions, guaranteeing that an immersed compact closed manifold $\Sigma^m\subset \R^n$ of class $C^1$ and of arbitrary dimension and codimension (or, more generally, an Ahlfors-regular compact…

In this paper, we mainly study the compactness and local structure of immersing surfaces in $\mathbb{R}^n$ with local uniform bounded area and small total curvature $\int_{\Sigma\cap B_1(0)} |A|^2$. A key ingredient is a new quantity which…

We study the volume functional on the space of constant scalar curvature metrics with a prescribed boundary metric. We derive a sufficient and necessary condition for a metric to be a critical point, and show that the only domains in space…

This paper continues the study of the ellipsoid embedding function of symplectic Hirzebruch surfaces parametrized by $b \in (0,1)$, the size of the symplectic blow-up. Cristofaro-Gardiner, et al. (arxiv: 2004.13062) found that if the…

We prove that if u is a bounded smooth function in the kernel of a nonnegative Schrodinger operator $-L=-(\Delta +q)$ on a parabolic Riemannian manifold M, then u is either identically zero or it has no zeros on M, and the linear space of…

This paper solves the rational noncommutative analog of Hilbert's 17th problem: if a noncommutative rational function is positive semidefinite on all tuples of hermitian matrices in its domain, then it is a sum of hermitian squares of…

Willmore surfaces are the extremals of the Willmore functional (possibly under a constraint on the conformal structure). With the characterization of Willmore surfaces by the (possibly perturbed) harmonicity of the mean curvature sphere…

A new version of the Hadwiger theorem on convex functions is established and an explicit representation of functional intrinsic volumes is found using new functional Cauchy-Kubota formulas. In addition, connections between functional…

The paper is centered around a new proof of the infinitesimal rigidity of smooth closed surfaces with everywhere positive Gauss curvature. We use a reformulation that replaces deformation of an embedding by deformation of the metric inside…

A new functional for simplicial surfaces is suggested. It is invariant with respect to Moebius transformations and is a discrete analogue of the Willmore functional. Minima of this functional are investigated. as an application a bending…

We define and study totally geodesic null hypersurfaces in Finsler spacetimes. We prove that the null convergence condition and a certain mild gravitational equation $\chi_\alpha=0$, imply the vanishing of the restriction of the Ricci…

A new conformally invariant energy for four-dimensional hypersurfaces is devised. It renders possible the study of a large class of curvature energies, and we show that their critical points are smooth. As corollaries, we obtain the…