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In this chapter we shall review some basic aspects of the theory of surfaces with constant mean curvature. Surfaces with constant mean curvature will arise as solutions of a variational problem associated to the area functional and related with the classical isoperimetric problem. We state the first and second variation formula for the area and we give the notion of stability of a cmc surface. Next, we introduce the complex analysis as a basic tool in the theory and this will allow to prove the Hopf theorem. Also, we compute the Laplacians of some functions that contain geometric information of a cmc surface. As the Laplacian operator is elliptic, the maximum principle yields height estimates of a graph of constant mean curvature. Finally, and with the aid of the expression of these Laplacians, we will derive the Barbosa-do Carmo theorem that characterizes a round sphere in the family of closed stable cmc surfaces of Euclidean space.

Let \(x:{\text M}^{n}\rightarrow\,{\text R}^{n+1}\) be an immersion of an orientable, n-dimensional manifold \({\text M}^{n}\) into the euclidean space \({\text R}^{n+1}\). The condition that x has nonzero constant mean curvature H = H 0 is known to be equivalent to the fact that xis a critical point of a variational problem.

In this text we outline the major techniques, concepts and results in mean curvature flow with a focus on higher codimension. In addition we include a few novel results and some material that cannot be found elsewhere.

Based on a natural approach for the time discretization of gradient flows a new time discretization for discrete Willmore flow of polygonal curves and triangulated surfaces is proposed. The approach is variational and takes into account an approximation of the L 2-distance between the surface at the current time step and the unknown surface at the new time step as well as a fully implicity approximation of the Willmore functional at the new time step. To evaluate the Willmore energy on the unknown surface of the next time step, we first ask for the solution of a inner, secondary variational problem describing a time step of mean curvature motion. The time discrete velocity deduced from the solution of the latter problem is regarded as an approximation of the mean curvature vector and enters the approximation of the actual Willmore functional. To solve the resulting nested variational problem in each time step numerically relaxation theory from PDE constraint optimization are taken into account. The approach is applied to polygonal curves and triangular surfaces and is independent of the co-dimension. Various numerical examples underline the stability of the new scheme, which enables time steps of the order of the spatial grid size.

The interest, in recent years, in the geometric processing of polygonal meshes, has spawned a whole range of algorithms to estimate curvature properties over smooth polygonal meshes. Being a discrete approximation of a C2 continuous surface, these methods attempt to estimate the curvature properties of the original surface. The best known methods are quite effective in estimating the total or Gaussian curvature but less so in estimating the mean curvature. In this work, we present a scheme to accurately estimate the mean curvature of smooth polygonal meshes using a one sided tube formula for the volume above the surface. In the presented comparison, the proposed scheme yielded results whose accuracy is amongst the highest compared to similar techniques for estimating the mean curvature.

A variational formulation of an image analysis problem has the nice feature that it is often easier to predict the effect of minimizing a certain energy functional than to interpret the corresponding Euler-Lagrange equations. For example, the equations of motion for an active contour usually contains a mean curvature term, which we know will regularizes the contour because mean curvature is the first variation of curve length, and shorter curves are typically smoother than longer ones. In some applications it may be worth considering Gaussian curvature as a regularizing term instead of mean curvature. The present paper provides a variational principle for this:We show that Gaussian curvature of a regular surface in three-dimensional Euclidean space is the first variation of an energy functional defined on the surface. Some properties of the corresponding motion by Gaussian curvature are pointed out, and a simple example is given, where minimization of this functional yields a nontrivial solution.

It is often advantageous in image processing and computer vision to consider images as surfaces imbedded in higher dimensional manifolds. It is therefore important to consider the theoretical and applied aspects of proper sampling of manifolds. We present a new sampling theorem for surfaces and higher dimensional manifolds. The core of the proof resides in triangulation results for manifolds with or without boundary, not necessarily compact. The proposed method adopts a geometric approach that is considered in the context of 2-dimensional manifolds (i.e surfaces), with direct applications in image processing. Implementations of these methods and theorems are illustrated and tested both on synthetic images and on real medical data.

A new vertex selection scheme using the curvature information of contour points for polygonal contour approximation is presented. The proposed method consists of two-step procedure. At first, we compute curvature values for contour points on the curvature scale space (CSS) and select high curvature points as principal vertices. Selected principal vertices, thereby, divide an overall contour into several contour segments. In the second step, the vertices for each contour segment are selected using the progressive vertex selection (PVS) method in order to obtain minimum number of vertices under the given maximum distance criterion (D*max).

Geometric evolution problems for curves and surfaces and especially curvature flow problems are an exciting and already classical mathematical research field. They lead to interesting systems of nonlinear partial differential equations and allow the appropriate mathematical modeling of physical processes such as material interface propagation, fluid free boundary motion, crystal growth.

Surfaces can be described locally and classified by their curvature values at every point. In this paper we investigate a grey-level based curvature estimator in combination with a sampling-error free integration technique of the curvature image. We compute shape descriptors as the bending energy and a global topological invariant, the Euler characterization. The integration of curvature values over the surface area is done by grey-volume integration. Our curvature estimator works on the orientation field of the surface, which does not require a segmentation of the surface. The estimated orientation fields has discontinuities mod π. It is mapped via the Knutsson mapping to a continuous representation in which the curvatures are computed.

5.1 S-Theory 5.2 The Mean Curvature Sphere 5.3 Hopf Fields 5.4 The Conformal Gauss Map

We present a new algorithm for computing discrete constant mean curvature surfaces in ℝ3. It is based on the definition of a discrete version of the conjugate surface construction for cmc surfaces. Here we solve a Plateau problem for a discrete minimal surface in S3 by computing a sequence of discrete harmonic maps F i : S3 → S3. The definition of a discrete conjugation allows to transform this sequence to a sequence of conjugate discrete maps which converges to a discrete cmc surface in ℝ3. The algorithm is applicable to free boundary value problems for cmc surfaces and led to the recent discovery of new compact cmc surfaces.