The study of surfaces with constant mean curvature (CMC) is one of the main topics in classical differential geometry. Moreover, CMC surfaces are important mathematical models for the physics of interfaces in the absence of gravity, where they separate two different media or for capillary : Capa dura. Loosely speaking, the curvature •of a curve at the point P is partially due to the fact that the curve itself is curved, and partially because the surface is curved. In order to somehow disentangle these two eﬁects, it it useful to deﬂne the two concepts normal curvature and geodesic curvature. We follow Kreyszig [14] in our Size: 1MB. Let ψ: M → N be a stable constant mean curvature immersion of a complete orientable surface into a three dimensional oriented Riemannian manifold. Then the following interesting, although partial results, are known Theorem 1. If N is compact and has positive Ricci curvature, then M is compact and connected, and genus(M)≤ 3. Theorem 2. minimal surfaces and surfaces of constant mean curvature. We discuss various types of natural Gauss images, the existence of principal curvatures, constant curvature sur-faces, Christoﬀel duality, Koenigs nets, contact element nets, s-isothermic nets, and interesting special cases such as discrete Delaunay surfaces derived from elliptic bil.

orienting such a surface with constant mean curvature H, we will assume H ≥0 andwillrefertothesurfaceasanH-surface. We next brieﬂy explain the contents of the three lectures in the course. The ﬁrst lecture introduced the notation, deﬁnitions and examples, as well as the ba-sic tools. In differential geometry, constant-mean-curvature (CMC) surfaces are surfaces with constant mean curvature. This includes minimal surfaces as a subset, but typically they are treated as special case.. Note that these surfaces are generally different from constant Gaussian curvature surfaces, with the important exception of the sphere. Biharmonic surfaces of constant mean curvature. We present a new method for modeling discrete constant mean curvature (CMC) surfaces, which arise frequently in nature and are highly demanded in architecture and other engineering applications. Our method is based on a novel use of the CVT (centroidal Voronoi tessellation) optimization : PanHao, ChoiYi-King, LiuYang, HuWenchao, DuQiang, PolthierKonrad, ZhangCaiming, WangWenping.

Bang-Yen Chen, in Handbook of Differential Geometry, Stability of surfaces with constant mean curvature. Since a compact constant mean curvature surface in E 3 is a critical point of the area functional with respect to volume-preserving normal variations, one can define the stability of such surfaces: A compact constant mean curvature surface in E 3 is called stable if A″(0) > 0. Constant mean curvature (CMC) surfaces have played a prominent role in di erential geometry. In , Charles Eug ene Delaunay introduced a way of constructing rotationally symmetric CMC surfaces in R3, by proving that a surface of revolution in R3 is a CMC surface if . complete, properly immersed, or embedded, surfaces of nonzero constant mean curvature in EV. Very little is known in this regard. For a long time, the only known examples of such surfaces were, besides the round sphere and the cylinder, a family of rotationally invariant surfaces discovered in by.