Related papers: Towards a constrained Willmore conjecture
We report about some results, interesting examples, problems and conjectures revolving around the parabolic Kostant partition functions, the parabolic Kostka polynomials and ``saturation'' properties of several generalizations of the…
The aim of the present article is to establish the connection between the existence of the limit along the normal and an admissible limit at a fixed boundary point for holomorphic functions of several complex variables.
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…
A proof of the Willmore conjecture is presented. With the help of the global Weierstrass representation the variational problem of the Willmore functional is transformed into a constrained variational problem on the moduli space of all…
The invariant theory for conformal hypersurfaces is studied by treating these as the conformal infinity of a conformally compact manifold: For a given conformal hypersurface embedding, a distinguished ambient metric is found (within its…
Several results about the union-closed sets conjecture are presented.
First introduced to describe surfaces embedded in $\mathbb{R}^3$, the Willmore invariant is a conformally-invariant extrinsic scalar curvature of a surface that vanishes when the surface minimizes bending and stretching. Both this invariant…
In this paper we discuss approximation of partially smooth functions. The problem arises naturally in the study of laminated currents.
The Willmore energy, alias bending energy or rigid string action, and its variation-the Willmore invariant-are important surface conformal invariants with applications ranging from cell membranes to the entanglement entropy in quantum…
We establish a conjecture of Mumford characterizing rationally connected complex projective manifolds in several cases.
We develop the calculus for hypersurface variations based on variation of the hypersurface defining function. This is used to show that the functional gradient of a new Willmore-like, conformal hypersurface energy agrees exactly with the…
We address a deep study of the convexity notions that arise in the study of weak* lower semicontinuity of supremal functionals as well as those raised by the power-law approximation of such functionals. Our quest is motivated by the…
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…
In this paper we consider surfaces which are critical points of the Willmore functional subject to constrained area. In the case of small area we calculate the corrections to the intrinsic geometry induced by the ambient curvature. These…
We formulate some problems and conjectures about semigroups of rational functions under composition. The considered problems arise in different contexts, but most of them are united by a certain relationship to the concept of amenability.
Beyond normal surfaces there are several open questions concerning 2- dimensional spaces. We present some results and conjectures along this line.
This paper surveys some selected topics in the theory of conformal metrics and their connections to complex analysis, partial differential equations and conformal differential geometry.
We survey recent developments on the Restriction conjecture.
The similarity between the Polya's conjecture and the Bonomol'nyi bound remind us to consider a physical approach to Polya's conjecture. We conjecture a duality between the waves and the soliton solutions on the surface. We consider the…
On the retention problem, we study the case of a functional constraint on the set of disturbances. A construction of resolving quasistrategy based on the method of programmed iterations is proposed.