1,721,005 research outputs found
On the structure of positive scalar curvature type graphs
No full textWe study some aspect of the pde satisfied by a hypersurface of R^{n+1} with positive scalar curvatur
On Properties of constant Mean Curvature Surfaces in H2xR
No full textWe discuss global properties of constant mean curvature surfaces
(H-surfaces) in H^2xR : maximum principle at infinity, halfspace type theorem,
non existence of simply connected surfaces with one end
A Survey on Alexandrov-Bernstein-Hopf Theorems
No full textWe give proofs of Alexandrov, Bernstein and Hopf Theorems. Then, we dis-
cuss the developments of the theory of constant mean curvature surfaces ensuing
from them
An Example of an Immersed Complete Genus One Minimal Surface in R^3 with Two Convex Ends
No full textWe prove the existence of a compact genus one immersed minimal surface M, whose boundary is the union of two immersed locally convex curves lying in parallel planes. M is a part of a complete minimal surface with two finite total curvature ends
The State of the Art of Bernstein's Problem
No full textWether the only minimal stable complete hypersurfaces in R^{n+1}, with 2< n < 8 are hyperplanes, is an open problem. We describe its historical motivations and our results obtained by exploring it
On The Existence and Uniqueness of Constant Mean Curvature Hypersurfaces in Hyperbolic Space
No full textWe prove that certain graphs of prescribed mean curvature in H^n
cannot have
an isolated singularity. Then we discuss a ux formula for surfaces with constant mean curvature in
H and some consequences of i
Some remarks on embedded hypersurfaces in hyperbolic space of constant curvature and spherical boundary
No full textWe consider embedded hypersurfaces M in hyperbolic space with compact boundary C and some r(th) mean curvature function H_r a positive constant. We investigate when symmetries of C are symmetries of M. We prove that if 0 less than or equal to H_r less than or equal to 1 and C is a sphere then M is a part of an equidistant sphere. For r = 1 (H_1 is the mean curvature) we obtain results when C is convex
On Hypersurfaces embedded in Euclidean Space with Positive Constant H_r Curvature
No full textWe consider hypersurfaces M embedded in a half-space R-+(n+1) with positive constant r(th) symmetric function of the principal curvatures
(H_r-surfaces). For such H_r-surfaces, 1 < r less than or equal to n, with strictly convex boundary in the boundary of R(+)(n+1) we show that, if H_r is small enough in terms of the geometry of the boundary of M, then M is topologically a disk. When r = 2, we also prove a compactness theorem for certain classes of H_2-surfaces
Stably Embedded Minimal Hypersurfaces
No full textWe use Schoen-Simon-Yau's curvature estimates to prove that the subfocal
tubular neighborhood of a non planar minimal hypersurface with bounded
second fundamental form, stably embedded in R^{n+1}, n < 5, whose radius
decays sufficiently slowly can not be embedded. In particular such hypersurfaces
admit no embedded tubular neighborhoods of constant radius, whatever
small the radius. However, assuming a further hypothesis on the embedding,
we prove that such hypersurfaces admit an embedded tube whose
radius decays sufficiently fast
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