scholarly journals The double cover relative to a convex domain and the relative isoperimetric inequality

2006 ◽  
Vol 80 (3) ◽  
pp. 375-382 ◽  
Author(s):  
Jaigyoung Choe
Keyword(s):  
Riemannian Manifold ◽  
Sectional Curvature ◽  
Isoperimetric Inequality ◽  
Convex Domain ◽  
Double Cover ◽  
Nonpositive Sectional Curvature ◽  
Relative Isoperimetric Inequality ◽  
Simply Connected

AbstractWe prove that a domain Ω in the exterior of a convex domain C in a four-dimensional simply connected Riemannian manifold of nonpositive sectional curvature satisfies the relative isoperimetric inequality 64π2 Vol(Ω)3 < Vol(∂Ω ~ ∂C)4. Equality holds if and only if Ω is an Euclidean half ball and ∂Ω ~ ∂C is a hemisphere.

The isoperimetric inequality for minimal surfaces in a Riemannian manifold

1999 ◽  
Vol 1999 (506) ◽  
pp. 205-214 ◽  
Author(s):  
Jaigyoung Choe
Keyword(s):  
Riemannian Manifold ◽  
Minimal Surface ◽  
Sectional Curvature ◽  
Isoperimetric Inequality ◽  
Gaussian Curvature ◽  
Minimal Surfaces ◽  
Constant Gaussian Curvature ◽  
Simply Connected

Abstract It is proved that every minimal surface with one or two boundary components in a simply connected Riemannian manifold with sectional curvature bounded above by a nonpositive constant K satisfies the sharp isoperimetric inequality 4π A ≦ L2 + K A2. Here equality holds if and only if the minimal surface is a geodesic disk in a surface of constant Gaussian curvature K.

scholarly journals AN INTRINSIC PROOF OF GROMOLL–GROVE DIAMETER RIGIDITY THEOREM

2007 ◽  
Vol 09 (03) ◽  
pp. 401-419 ◽  
Author(s):  
JIANGUO CAO ◽  
HONGYAN TANG
Keyword(s):  
Riemannian Manifold ◽  
Sectional Curvature ◽  
Projective Spaces ◽  
Rigidity Theorem ◽  
Simply Connected

Using the spherical trip theorem, we present a new intrinsic proof of Gromoll–Grove diameter rigidity theorem: "If a simply-connected Riemannian manifold has sectional curvature ≥ 1 and diameter [Formula: see text], then either it is homeomorphic to a sphere, or it is isometric to one of classic projective spaces".

scholarly journals An integral formula for hypersurfaces in space forms

1995 ◽  
Vol 37 (3) ◽  
pp. 337-341 ◽  
Author(s):  
Theodoros Vlachos
Keyword(s):  
Riemannian Manifold ◽  
Sectional Curvature ◽  
Distance Function ◽  
Integral Formula ◽  
Constant Sectional Curvature ◽  
Position Vector ◽  
Space Forms ◽  
Image Position ◽  
Simply Connected

Let be an n+ 1-dimensional, complete simply connected Riemannian manifold of constant sectional curvature c and We consider the function r(·) = d(·, P0) where d stands for the distance function in and we denote by grad r the gradient of The position vector (see [1]) with origin P0 is defined as where ϕ(r)equalsr, if c = 0, c< 0 or c <0 respectively.

scholarly journals Perturbations of the Harmonic Map Equation

2003 ◽  
Vol 05 (04) ◽  
pp. 629-669 ◽  
Author(s):  
T. Kappeler ◽  
S. Kuksin ◽  
V. Schroeder
Keyword(s):  
Riemannian Manifold ◽  
Sectional Curvature ◽  
Homotopy Class ◽  
Harmonic Map ◽  
Poincaré Inequality ◽  
Classical Solutions ◽  
Important Ingredient ◽  
Closed Riemannian Manifold ◽  
Nonpositive Sectional Curvature ◽  
Target Manifold

We consider perturbations of the harmonic map equation in the case where the target manifold is a closed Riemannian manifold of nonpositive sectional curvature. For any semilinear and, under some extra conditions, quasilinear perturbation, the space of classical solutions within a homotopy class is proved to be compact. An important ingredient for our analysis is a new inequality for maps in a given homotopy class which can be viewed as a version of the Poincaré inequality for such maps.

scholarly journals The diameter of an immersed Riemannian manifold with bounded mean curvature

1981 ◽  
Vol 31 (2) ◽  
pp. 189-192 ◽  
Author(s):  
Th. Koufogiorgos ◽  
Ch. Baikoussis
Keyword(s):  
Riemannian Manifold ◽  
Isometric Immersion ◽  
Mean Curvature ◽  
Curvature Vector ◽  
Complete Riemannian Manifold ◽  
Mean Curvature Vector ◽  
Nonpositive Sectional Curvature ◽  
The Mean ◽  
Simply Connected ◽  
Bounded Norm

AbstractLet M be an n-dimensional complete Riemannian manifold with Ricci curvature bounded from below. Let be an N-dimensional (N < n) complete, simply connected Riemannian manifold with nonpositive sectional curvature. We shall prove in this note that if there exists an isometric immersion φ of M into with the property that the immersed manifold is contained in a ball of radius R and that the mean curvature vector H of the immersion has bounded norm ∥H∥ > H0, (H0 > 0) then R > H−10.

scholarly journals Some results on stable p-harmonic maps

1994 ◽  
Vol 36 (1) ◽  
pp. 77-80 ◽  
Author(s):  
Leung-Fu Cheung ◽  
Pui-Fai Leung
Keyword(s):  
Riemannian Manifold ◽  
Critical Point ◽  
Harmonic Maps ◽  
Harmonic Map ◽  
Energy Functional ◽  
Complete Riemannian Manifold ◽  
Image Position ◽  
Simply Connected

For each p ∈ [2, ∞)a p-harmonic map f:Mm→Nn is a critical point of the p-energy functionalwhere Mm is a compact and Nn a complete Riemannian manifold of dimensions m and n respectively. In a recent paper [3], Takeuchi has proved that for a certain class of simply-connected δ-pinched Nn and certain type of hypersurface Nn in ℝn+1, the only stable p-harmonic maps for any compact Mm are the constant maps. Our purpose in this note is to establish the following theorem which complements Takeuchi's results.

TOPONOGOV-TYPE AREA COMPARISON THEOREM OF 2-DIMENSIONAL MANIFOLDS

2013 ◽  
Vol 15 (03) ◽  
pp. 1350007
Author(s):  
XIAOLE SU ◽  
HONGWEI SUN ◽  
YUSHENG WANG
Keyword(s):  
Riemannian Manifold ◽  
Comparison Theorem ◽  
Constant Curvature ◽  
Dimensional Manifold ◽  
Geodesic Triangle ◽  
Type Area ◽  
Simply Connected

Let △p1p2p3 be a geodesic triangle on M, a complete 2-dimensional Riemannian manifold of curvature ≥ k, and let [Formula: see text] be its comparison triangle on [Formula: see text] (a complete and simply connected 2-dimensional manifold of constant curvature k). Our main result is that if △p1p2p3 is areable, then its area is not less than that of [Formula: see text].

scholarly journals Hyperbolic rank rigidity for manifolds of -pinched negative curvature

2018 ◽  
Vol 40 (5) ◽  
pp. 1194-1216
Author(s):  
CHRIS CONNELL ◽  
THANG NGUYEN ◽  
RALF SPATZIER
Keyword(s):  
Riemannian Manifold ◽  
Symmetric Space ◽  
Sectional Curvature ◽  
Negative Curvature ◽  
Jacobi Field ◽  
Rigidity Result ◽  
Locally Symmetric Space ◽  
Rank One ◽  
Sectional Curvatures

A Riemannian manifold $M$ has higher hyperbolic rank if every geodesic has a perpendicular Jacobi field making sectional curvature $-1$ with the geodesic. If, in addition, the sectional curvatures of $M$ lie in the interval $[-1,-\frac{1}{4}]$ and $M$ is closed, we show that $M$ is a locally symmetric space of rank one. This partially extends work by Constantine using completely different methods. It is also a partial counterpart to Hamenstädt’s hyperbolic rank rigidity result for sectional curvatures $\leq -1$, and complements well-known results on Euclidean and spherical rank rigidity.

scholarly journals Positively curved 6-manifolds with simple symmetry groups

2002 ◽  
Vol 74 (4) ◽  
pp. 589-597 ◽  
Author(s):  
FUQUAN FANG
Keyword(s):  
Sectional Curvature ◽  
Isometry Group ◽  
Symmetry Groups ◽  
Positive Sectional Curvature ◽  
Identity Component ◽  
Simply Connected ◽  
Lie Subgroup ◽  
Positively Curved

Let M be a simply connected compact 6-manifold of positive sectional curvature. If the identity component of the isometry group contains a simple Lie subgroup, we prove that M is diffeomorphic to one of the five manifolds listed in Theorem A.

Sign in / Sign up

Export Citation Format

Share Document