We provide a new topological obstruction for complete stable minimal hypersurfaces in IR n+1. For n ≥ 3, we prove that a complete orientable stable minimal hypersurface in IR n+1 cannot have more than one end by showing the existence of a bounded harmonic function based on the Sobolev inequality for minimal submanifolds [MS] and by applying the Liouville… (More)
O ur goal is to em bed free-form constraints into a graphical m odel. W ith such constraints a graphic can m aintain its visual integrity— and break rules tastefully— while being m anipulated by a casualuser. A typicalparam eterized graphic does notm eet these needs because its configuration space contains nonsense im ages in m uch higher… (More)
We survey comparison results that assume a bound on the man-ifold's Ricci curvature.
We obtain a gradient estimate for the Gauss maps from complete spacelike constant mean curvature hypersurfaces in Minkowski space into the hyperbolic space. As applications, 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… (More)
Using an analogue of Myers' theorem for minimal surfaces and three dimensional topology, we prove the diameter sphere theorem for Ricci curvature in dimension three and a corresponding eigenvalue pinching theorem. This settles these two problems for closed manifolds with positive Ricci curvature since they are both false in dimensions greater than three.… (More)
Paper presentations are coming up, and we talked about how to give a good talk. Here is the slide deck I showed, which has a lot of other good references: Which I used in part as an excuse to cover a backdrop of work leading up to this result. A classic historical graphic reference (1976!) is the following: Burtnyk, Nester, and Marceli Wein. "Interactive… (More)
This thesis is based on the following papers, which are referred to in the text by their Roman numerals. Reprints were made with permission from the respective publishers. Author Contribution The author wishes to clarify his contributions to the included papers. I Performed a large part of the calculations and contributed partly to manuscript writing. II… (More)
Using the Green's function and some comparison theorems, we obtain a lower bound on the rst 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… (More)