See for these results apart from 6, other equalities and estimates have been obtained see, which are universal in the sense that they refer to all surfaces of. D3valued holomorphic 1form on a suitable domain in c in the case of spacelike maximal surfaces resp. In order to discuss the mean curvature of a surface, some restrictions need to be made on the surfaces being used. In these cases, hydrolysis is shown to take place at the hole edges where the surface curvature is high, providing an exposed site for hydrolysis. The curvatures of all planar surfaces were estimated correctly as being zero. A surface of gaussian curvature zero is locally isometric to the plane, and is said to. Bicomplex extensions of zero mean curvature surfaces in r. Minimal surfaces are defined as surfaces with zero mean curvature.
One of our main results states that those surfaces have nonpositive gaussian curvature in r3. In relativity theory there is a connection between the distribution of mass and energy more precisely, between the energymomentum tensor and the curvature of spacetime. Having a very slight curvature in the vertical dimension and not in the horizontal, since that would screw with the hinge mechanism makes perfect sense. Surfaces with zero mean curvature vector in 4dimensional. In this book, the author presents numerous examples of constant. Timelike surfaces with zero mean curvature in minkowski 4space. We will obtain the following theorem for a spacelike surface in n4 1 with zero mean curvature vector.
The curvature indicates how the surface orientation changes as one traverses the surface. It is a straightforward matter to compute the gaussian and mean curvature of a ruled surface. By calabis correspondence, this also gives a family of explicit selfsimilar solutions for the minimal surface. Jellet showed that if is a compact starshaped surface in with constant mean curvature, then it is the standard sphere. Pdf parametric surfaces of prescribed mean curvature. A surface is a minimal surface if and only if the mean curvature is zero. The mean curvature and conformality hot network questions an image with lower resolution is of higher file size while that of a higher resolution is of lower file size.
The integral of the mean curvature times the normal vector of an arbitrary. Mean curvature has a special physical signification. Negative curvature, surface of encyclopedia of mathematics. It relies on an edgewise parallel pair m,s of polyhedral surfaces. Zero mean curvature surfaces in l3 containing a lightlike line. I want to calculate the curvature in a different manner. The focus of study is a timeparameterized family of k dimensional surfaces in the n dimensional euclidean space 1. This book explains the notion of brakkes mean curvature flow and its existence and regularity theories without assuming familiarity with geometric measure theory. I want to make cuts that pass through the center, and evaluate the curvature. In this paper, we consider the existence of selfsimilar solution for a class of zero mean curvature equations including the borninfeld equation, the membrane equation and maximal surface equation. A minimal surface is a surface which has zero mean curvature at all points. The mean curvature at a point on a surface is the average of the principal curvatures at the point i.
Minimal surfaces are classically defined as surfaces of zero mean curvature in. That is to say, the surface is shrinking by homothety. This is equivalent to having zero mean curvature see definitions below. A curvature theory for discrete surfaces based on mesh. In a number of instances this surface has been considered to be minimal and periodic corresponding to zero mean curvature at every point luzzati et al 1996. During the 19th century and beginning of the 20th century, the desire to give substance to geometric ideas led to the creation of mathematical models, envisioned by. In particular, we obtain all timelike surfaces of revolution with constant mean curvature. Nonconvex anisotropic surface energy and zero mean.
A surface whose mean curvature is constant but nonzero is obtained when we try to minimize the area of a closed surface without changing the volume it encloses. By this formula, each zero mean curvature surface is presented locally 1. In 1841 delaunay proved that the only surfaces of revolution with constant mean curvature were the surfaces obtained by rotating the roulettes of the conics. Surfaces with constant mean curvature katsuei kenmotsu. This is similar to the surface formed by an open soap film suspended on a wire frame where the pressure is equal on both sides of the film and hence the mean curvature at every point is zero. Finding a minimal surface of a boundary with specified constraints is a problem in the.
The mean curvature of a surface at a point is one half the sum of the principal curvatures at that point. Mathematical models newton was the forerunner in the pursuit of various mathematical principles, and many fields of natural sciences have since followed in his footsteps. Weisstein, curvature a modern day application of curvature can be found in the study of modern physics. When the mean curvature is a non zero constant, we prove that if the surface is foliated by circles in parallel planes, then it must be rotational. Surfaces with constant mean curvature ams bookstore. Im no expert in this field, but i am familiar with the classification of rotationally symmetric surfaces with constant mean curvature by delaunay. Lectures on mean curvature flow and related equations. General description of surface with zero gaussian curvature. A surface is minimal provided its mean curvature is zero everywhere. Surfaces with zero mean curvature everywhere are minimal surfaces. More recently, besl used the mean and gaussian curvature signs calculated from range data to produce a similar taxonomy, only with greater differentiation of the hyperboloidal surfaces.
We know that ellipsoids and hyperboloids are surfaces of revolution provided that two of their axes are equal. Furthermore, a surface which evolves under the mean curvature of the surface, is said to obey a heattype equation called the mean curvature flow equation. Most surfaces will contain regions of positive gaussian curvature elliptical points and regions of negative gaussian curvature separated by a curve of points with zero gaussian curvature called a parabolic line. If the surface n begins to move by its mean curvature vector, this produces the same normal velocity as moving by the vector x2. The gaussian curvature is the product of those values. These are the plane, cylinder, sphere, the catenoid, the unduloid and nodoid in 1853 j. Surface curvatures, has one principal curvature equal to zero and the other equal to the inverse of the radius of its cross section. Nonconvex anisotropic surface energy and zero mean curvature surfaces in the lorentzminkowski space atsufumi honda, miyuki koiso and yasuhiro tanaka received on march 7, 20 revised on april 8, 20 abstract. Therefore the curvature is the derivative of the surface normal, and thus the second derivative of position. Any point with zero mean curvature has negative or zero gaussian curvature. Fast estimation of mean curvature on the surface of a 3d. The gaussian curvature signifies a peak, a valley, or a saddle point, depending on the sign. Assume u is a smooth minimizer of the area integral iw.
Mean curvature the surfaces in the project possess vanishing mean curvature. On any timelike surface with zero mean curvature in the four dimensional minkowski space we introduce special geometric. It is well known that spacelike maximal surfaces and timelike minimal surfaces in lorentzminkowski 3space l 3 have singularities i. A complete surface of gaussian curvature zero in euclidean three space is a cylinder where a cylinder means the surface generated by the lines parallel to a given axis passing through a fixed curve in the subspace perpendicular to the axis. There is a rich and wellknown theory of minimal surfaces. We consider the general case of spacelike surfaces with the same mean curvature with respect to both metrics. Prove that graph of u is a surface of constant mean curvature. Mesoporous inorganic phases templated by surfactant or block.
We will show in this paper that if the initial surface satis. An anisotropic surface energy functional is the integral of an energy density function over a surface. By considering 6 and 18, a rotational surface m 1b has zero mean curvature if. This includes minimal surfaces as a subset, but typically they are treated as special case. A curvature theory for discrete surfaces based on mesh parallelity 3 a discrete version of this construction is of central importance for this paper. Classic examples include the catenoid, helicoid and enneper surface. Explicit selfsimilar solutions for a class of zero mean. A surface whose mean curvature is zero at each point is a minimal surface, and it is known that such surfaces are models for soap film. If positive, a valley or peak, if negative, a saddle point, and if the gaussian curvature is zero, than the surface is flat in at least one direction.
The directions in the tangent plane for which takes maximum and minimum values are. The mean curvature of the surface of a liquid is related to the capillary effect. Surfaces of revolution and constant curvature surfaces of revolution form the most easily recognized class of surfaces. There are three classes of such surfaces, the least obvious but most interesting being the class of tangent developables. It turns out that ss is the eulerlagrange equation of a.
A surface whose mean curvature is zero at each point is a minimal surface. This partial differential equation is the minimal surface equation. Introduction books on the classical differential geometry of surfaces in 3space usually prove a theorem to the effect that a surface of gaussian curvature 0 is a. Therefore it has become customary to call a surface with zero mean curvature a minimal surface. We adopt similar notation to huisken in hul and hu2. In mathematics, a minimal surface is a surface that locally minimizes its area. Is there a complete classification of constant mean. Gaussian curvature is positive for surfaces like a hemisphere, which are either concave or convex in all the directions. The mean curvature of a surface is an extrinsic parameter measuring how the surface is curved in the threedimensional space.
For example, if v is a ksphere with radius r centered at the origin with unit magnitude surface tension, then hv. One way to examine how much a surface bends is to look at the curvature of curves on the surface. In mathematics, the mean curvature h \displaystyle h h of a surface s \ displaystyle s s. In differential geometry, constant mean curvature cmc surfaces are surfaces with constant mean curvature. Hopf conjectured that this, in fact, holds for all compact immersions. The expanded surface presents some irregularities in the topography of the surface, which i want to evaluate by measuring the curvature. These repeatabilities were significant, except mean. Having a perfectly flat screen surface on a device this size is actually not a good thing since it would make it much easier to scratch the screen if you were to lay it face down. Geometric properties of surfaces with the same mean. Let s be an immersion of an oriented, compact hypersurface with constant mean curvature h 6 0 in r n. In the special case where the surface tension is isotropic, i. However see, in the case of there is a counterexample in the form of a weakly irregular surface.
356 1017 748 754 298 452 732 881 101 1213 667 466 435 1029 705 339 1103 1249 636 1237 62 634 1468 825 193 344 1471 953 106 1269 636 1320 392 1025 867 969 1525 1166 1126 110 101 1187 313 1001 735 1492 1497 1287 1218 345