A new Poincaré type inequality

Using the Green's function and some comparison theorems, we obtain a lower bound on the first Dirichlet eigenvalue for a domain D on a complete manifold with curvature bounded from above. And the lower bound is given explicitly in terms of the diameter of D and the dimension of D. This result can be considered as an analogue for nonpositively curved manifolds of Li-Schoen [L-Sc] and Li-Yau's [L-Ya] theorems for nonnegatively curved manifolds. We also give conditions under which a minimal hypersurface is stable in spaces with constant curvature. 1 The Main Theorem The Poincar'e inequality is one of the fundamental inequalities in the study of partial differential equations. It is one of the essential tools for the derivation of many of the a priori estimates for the solutions of the differential equations. In this paper, we will derive a Poincar'e inequality for domains in nonpositively curved manifolds and in minimal submanifolds of a nonpositively curved manifold. 1 1991 Mathemati..

A Bernstein theorem for complete spacelike constant mean curvature hypersurfaces in Minkowski space

We obtain a gradient estimate for the Gauss maps from complete spacelike constant mean curvature hypersurfaces in Minkowski space into the hyperbolic space. As an application, we prove a Bernstein theorem which says that if the image of the Gauss map is bounded from one side, then the spacelike constant mean curvature hypersurface must be linear. This result extends the previous theorems obtained by B. Palmer [Pa] and Y.L. Xin [Xin1] where they assume that the image of the Gauss map is bounded. We also prove a Bernstein theorem for spacelike complete surfaces with parallel mean curvature vector in four-dimensional spaces. 1 Introduction The classical Bernstein theorem [B] states that the only entire solutions of the minimal hypersurface equation in Euclidean space IR n+1 n X i=1 @ @x i ( @u @x i q 1 + jruj 2 ) = 0 (1) 1 1991 Mathematics Subject Classification. Primary 53C21, 53C42. 2 Key words and phrases. Spacelike hypersurfaces, constant mean curvature, harmonic maps, g..

Abstract Accessible Animation and Customizable Graphics via Simplicial Configuration Modeling

Our goal is to embed free-form constraints into a graphical model. With such constraints a graphic can maintain its visual integrity—and break rules tastefully—while being manipulated by a casual user. A typical parameterized graphic does not meet these needs because its configuration space contains nonsense images in much higher proportion than desirable images, and the casual user is apt to ruin the graphic on any attempt to modify or animate it. We therefore model the small subset of a given graphic’s configuration space that maps to desirable images. In our solution, the basic building block is a simplicial complex—the most practical data structure able to accommodate the variety of topologies that can arise. The configuration-space model can be built from a cross product of such complexes. We describe how to define the mapping from this space to the image space. We show how to invert that mapping, allowing the user to manipulate the image without understanding the structure of the configuration-space model. We also show how to extend the mapping when the original parameterization contains hierarchy, coordinate transformations, and other nonlinearities. Our software implementation applies simplicial configuration modeling to 2D vector graphics