Related papers: The Euler-Helfrich Functional
We establish the existence and uniqueness of some smooth accelerating transonic flows governed by the three dimensional steady compressible Euler equations with an external force in cylinders with arbitrary cross sections, which include…
This is a companion paper to arXiv:1207.3529 where we introduced the spinorial energy functional and studied its main properties in dimensions equal or greater than three. In this article we focus on the surface case. A salient feature here…
We prove a monotonicity identity for compact surfaces with free boundaries inside the boundary of unit ball in $\mathbb R^n$ that have square integrable mean curvature. As one consequence we obtain a Li-Yau type inequality in this setting,…
We consider the extrinsic geometry of surfaces in simply isotropic space, a three-dimensional space equipped with a rank 2 metric of index zero. Since the metric is degenerate, a surface normal cannot be unequivocally defined based on…
We consider Lie minimal surfaces, the critical points of the simplest Lie sphere invariant energy, in Riemannian space forms. These surfaces can be characterized via their Euler-Lagrange equations, which take the form of differential…
We study a three-dimensional incompressible viscous fluid in a horizontally periodic domain with finite depth whose free boundary is the graph of a function. The fluid is subject to gravity and generalized forces arising from a surface…
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…
In this paper, we study Hamiltonian stationary Lagrangian surfaces in complex space forms. We first show that when the mean curvature is a non-zero constant, the second fundamental form is parallel. We then consider the case in which the…
Consider a closed lipid membrane (vesicle), modeled as a two-dimensional surface, described by a geometrical hamiltonian that depends on its extrinsic curvature. The vanishing of its first variation determines the equilibrium configurations…
By means of Surface Evolver (Exp. Math,1,141 1992), a software package of brute-force energy minimization over a triangulated surface developed by the geometry center of University of Minnesota, we have numerically searched the…
We consider nematic liquid crystals in a bounded, convex polyhedron described by a director field n(r) subject to tangent boundary conditions. We derive lower bounds for the one-constant elastic energy in terms of topological invariants.…
The discrete model of the real symmetric one-matrix ensemble is analyzed with a cubic interaction. The partition function is found to satisfy a recursion relation that solves the model. The double-scaling limit of the recursion relation…
We provide a novel existence result for energy-variational solutions to a general class of evolutionary partial differential equations. Compared to previous works on this solution concept, the generalization is mainly twofold: a relaxation…
In this paper we mainly study the type II singularities of the mean curvature flow from a symplectic surface or from an almost calibrated Lagrangian surface in a K \"ahler-Einstein surface. We show the relation between the maximum of the…
Motivated by the power of subregion/subregion duality for constraining the bulk geometry in gauge/gravity duality, we pursue a comprehensive and systematic approach to the behavior of extremal surfaces under perturbations. Specifically, we…
We determine the Hilbert-Kunz function of plane elliptic curves in odd characteristic, as well as over arbitrary fields the generalized Hilbert-Kunz functions of nodal cubic curves. Together with results of K. Pardue and P. Monsky, this…
We study the free boundary Euler equations with surface tension in three spatial dimensions, showing that the equations are well-posed if the coefficient of surface tension is positive. Then we prove that under natural assumptions, the…
A statistical method for calculating equilibrium solutions of the shallow water equations, a model of essentially 2-d fluid flow with a free surface, is described. The model contains a competing acoustic turbulent {\it direct} energy…
We investigate the existence, uniqueness, and $L^1$-contractivity of weak solutions to a porous medium equation with fractional diffusion on an evolving hypersurface. To settle the existence, we reformulate the equation as a local problem…
We study the global influence of curvature on the free energy landscape of two-dimensional binary mixtures confined on closed surfaces. Starting from a generic effective free energy, constructed on the basis of symmetry considerations and…