相关论文: A conformal energy for simplicial surfaces
The Willmore energy plays a central role in the conformal geometry of surfaces in the conformal 3-sphere \(S^3\). It also arises as the leading term in variational problems ranging from black holes, to elasticity, and cell biology. In the…
We introduce a smooth quadratic conformal functional and its weighted version $$W_2=\sum_e \beta^2(e)\quad W_{2,w}=\sum_e (n_i+n_j)\beta^2(e),$$ where $\beta(e)$ is the extrinsic intersection angle of the circumcircles of the triangles of…
We prove that a certain discrete energy for triangulated surfaces, defined in the spirit of discrete differential geometry, converges to the Willmore energy in the sense of $\Gamma$-convergence. Variants of this discrete energy have been…
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 propose the study of a conformally invariant functional for surfaces of complex projective plane which is closely related to the classical Willmore functional. We show that minimal surfaces of complex projective plane are critical for…
This is the first comprehensive introduction to the authors' recent attempts toward a better understanding of the global concepts behind spinor representations of surfaces in 3-space. The important new aspect is a quaternionic-valued…
In the search for appropriate discretizations of surface theory it is crucial to preserve such fundamental properties of surfaces as their invariance with respect to transformation groups. We discuss discretizations based on M\"obius…
Recently a significant interest in ferromagnetic curved thin films has appeared. In particular, thin spherical shells are currently of great interest due to their capability to support skyrmion solutions which can be stabilized by curvature…
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…
Continuous conformal transformation minimizes the conformal energy. The convergence of minimizing discrete conformal energy when the discrete mesh size tends to zero is an open problem. This paper addresses this problem via a careful error…
We develop a bubble tree construction and prove compactness results for $W^{2,2}$ branched conformal immersions of closed Riemann surfaces, with varying conformal structures whose limit may degenerate, in a compact Riemannian manifold with…
Functionals involving surface curvature are important across a range of scientific disciplines, and their extrema are representative of physically meaningful objects such as atomic lattices and biomembranes. Inspired in particular by the…
The Willmore energy of a closed surface in R^n is the integral of its squared mean curvature, and is invariant uner M\"obius transformations of R^n. We show that any torus in R^3 with energy at most $8 \pi-delta$ has a representative under…
The theory of string-like continuous curves and discrete chains have numerous important physical applications. Here we develop a general geometrical approach, to systematically derive Hamiltonian energy functions for these objects. In the…
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…
In a recent paper by Iglesias, Rumpf and Scherzer (Found. Comput. Math. 18(4), 2018) a variational model for deformations matching a pair of shapes given as level set functions was proposed. Its main feature is the presence of anisotropic…
We show that the quantization of energy for Willmore spheres into closed Riemannian manifolds holds provided that the Willmore energy and the area are uniformly bounded. The analogous energy quantization result holds for Willmore surfaces…
The most general conformally invariant bending energy of a closed four-dimensional surface, polynomial in the extrinsic curvature and its derivatives, is constructed. This invariance manifests itself as a set of constraints on the…
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…
This paper is dedicated to the exploration of the conformal Willmore functional for surfaces within 4-dimensional conformal manifolds. We provide a detailed calculation of both the first and second variations, and present the Euler-Lagrange…