Toponogov's Theorem Research Articles

Min‐Oo Conjecture for Fully Nonlinear Conformally Invariant Equations

AbstractIn this paper we show rigidity results for supersolutions to fully nonlinear, elliptic, conformally invariant equations on subdomains of the standard n‐sphere under suitable conditions along the boundary. We emphasize that our results do not assume concavity on the fully nonlinear equations we will work with.This proves rigidity for compact, connected, locally conformally flat manifolds (M, g) with boundary such that the eigenvalues of the Schouten tensor satisfy a fully nonlinear elliptic inequality and whose boundary is isometric to a geodesic sphere ∂D(r), where D(r) denotes a geodesic ball of radius r ∈ (0, π/2] in , and totally umbilical with mean curvature bounded below by the mean curvature of this geodesic sphere. Under the above conditions, (M, g) must be isometric to the closed geodesic ball .As a side product, in dimension 2 our methods provide a new proof to Toponogov's theorem about rigidity of compact surfaces carrying a shortest simple geodesic. Roughly speaking, Toponogov's theorem is equivalent to a rigidity theorem for spherical caps in the hyperbolic three‐space ℍ3. In fact, we extend it to obtain rigidity for supersolutions to certain Monge‐Ampère equations. © 2019 Wiley Periodicals, Inc.

Improved Critical Eigenfunction Restriction Estimates on Riemannian Surfaces with Nonpositive Curvature

We show that one can obtain improved L 4 geodesic restriction estimates for eigenfunctions on compact Riemannian surfaces with nonpositive curvature. We achieve this by adapting Sogge’s strategy in (Improved critical eigenfunction estimates on manifolds of nonpositive curvature, Preprint). We first combine the improved L 2 restriction estimate of Blair and Sogge (Concerning Toponogov's Theorem and logarithmic improvement of estimates of eigenfunctions, Preprint) and the classical improved $${L^\infty}$$ estimate of Bérard to obtain an improved weak-type L 4 restriction estimate. We then upgrade this weak estimate to a strong one by using the improved Lorentz space estimate of Bak and Seeger (Math Res Lett 18(4):767–781, 2011). This estimate improves the L 4 restriction estimate of Burq et al. (Duke Math J 138:445–486, 2007) and Hu (Forum Math 6:1021–1052, 2009) by a power of $${(\log\log\lambda)^{-1}}$$ . Moreover, in the case of compact hyperbolic surfaces, we obtain further improvements in terms of $${(\log\lambda)^{-1}}$$ by applying the ideas from (Chen and Sogge, Commun Math Phys 329(3):435–459, 2014) and (Blair and Sogge, Concerning Toponogov's Theorem and logarithmic improvement of estimates of eigenfunctions, Preprint). We are able to compute various constants that appeared in (Chen and Sogge, Commun Math Phys 329(3):435–459, 2014) explicitly, by proving detailed oscillatory integral estimates and lifting calculations to the universal cover $${\mathbb{H}^2}$$ .

On Toponogov's comparison theorem for Alexandrov spaces

In this expository note, we present a transparent proof of Toponogov's theorem for Alexandrov spaces in the general case, not assuming local compactness of the underlying metric space. More precisely, we show that if M is a complete geodesic metric space such that the Alexandrov triangle comparisons for curvature greater than or equal to k are satisfied locally, then these comparisons also hold in the large. The proof is a modification of an argument due to Plaut.

Manifolds with asymptotically nonnegative minimal radial curvature

Abstract In this paper we extend results on the geometry of manifolds with asymptotically nonnegative curvature to manifolds with asymptotically nonnegative minimal radial curvature, showing that most of the results obtained by [U. Abresch, Lower curvature bounds, Toponogov's theorem, and bounded topology. Ann. Sci. École Norm. Sup. (4) 18 (1985), 651–670. MR839689 (87j:53058) Zbl 0595.53043], [A. Kasue, A compactification of a manifold with asymptotically nonnegative curvature. Ann. Sci. École Norm. Sup. (4) 21 (1988), 593–622. MR982335 (90d:53049) Zbl 0662.53032] and [S.-H. Zhu, A volume comparison theorem for manifolds with asymptotically nonnegative curvature and its applications. Amer. J. Math. 116 (1994), 669–682. MR1277451 (95c:53049) Zbl 0953.53027] hold in a more general context. Particularly, we show that there exists one and only one tangent cone at infinity to each such manifold, in contrast with the class of manifolds of nonnegative Ricci curvature.

Open manifolds with asymptotically nonnegative curvature

In this paper we establish a generalization of Toponogov's theorem for manifolds with asymptotically nonnegative sectional curvature, and we give a pinching condition under which asymptotically nonnegative curved manifolds are diffeomorphic to $\mathbb{R}^{n}$.

Toponogov's triangle comparison theorem in model spaces of nonconstant curvature

So far there exist two versions of Toponogov's triangle comparison theorem with surfaces of revolution as model spaces. U. Abresch has considered surfaces of nonpositive curvature and D. Elerath has investigated embedded surfaces of positive curvature. As the classical theorem, these two extensions are proved applying Rauch's comparison theorems. In this paper we present a new version of Toponogov's theorem generalizing all comparison theorems mentioned above. The new proof is based on estimates for the second fundamental tensor of distance spheres and handles the rigidity as well.

Lower curvature bounds, Toponogov's theorem, and bounded topology. II

Lower curvature bounds, Toponogov's theorem, and bounded topology. II

Lower curvature bounds, Toponogov's theorem, and bounded topology

Lower curvature bounds, Toponogov's theorem, and bounded topology