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We prove integral curvature bounds in terms of the Betti numbers for compact submanifolds of the Euclidean space with low codimension. As an application, we obtain topological obstructions for \(\delta \)-pinched immersions. Furthermore, we obtain intrinsic obstructions for minimal submanifolds in spheres with pinched second fundamental form. 相似文献
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We find new obstructions to the existence of complete Riemannian metric of nonnegative sectional curvature on manifolds with
infinite fundamental groups. In particular, we construct many examples of vector bundles whose total spaces admit no nonnegatively
curved metrics.
Received February 11, 2000 / Published online February 5, 2001 相似文献
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Keomkyo Seo 《Monatshefte für Mathematik》2012,97(3):525-542
In this paper, we provide various Sobolev-type inequalities for smooth nonnegative functions with compact support on a submanifold with variable mean curvature in a Riemannian manifold whose sectional curvature is bounded above by a constant. We further obtain the corresponding linear isoperimetric inequalities involving mean curvature. We also provide various first Dirichlet eigenvalue estimates for submanifolds with bounded mean curvature. 相似文献
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Keomkyo Seo 《Monatshefte für Mathematik》2012,166(3-4):525-542
In this paper, we provide various Sobolev-type inequalities for smooth nonnegative functions with compact support on a submanifold with variable mean curvature in a Riemannian manifold whose sectional curvature is bounded above by a constant. We further obtain the corresponding linear isoperimetric inequalities involving mean curvature. We also provide various first Dirichlet eigenvalue estimates for submanifolds with bounded mean curvature. 相似文献
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Xu Hongwei Han Wei 《高校应用数学学报(英文版)》2005,20(4):475-482
§1Introduction LetMnbeann-dimensionalcompactRiemannianmanifoldisometricallyimmersedinto an(n+p)-dimentionalcompleteandsimplyconnectedRiemannianmanifoldFn+p(c)with constantcurvaturec.DenotebyKMandHthesectionalcurvatureandmeancurvatureofM respectively.In[10],Yauprovedthefollowingstrikingresult.TheoremA.LetMnbeann-dimensionalorientedcompactminimalsubmanifoldin Sn+p(1).IfthesectionalcurvatureofMisnotlessthanp-12p-1,thenMiseitherthetotally geodesicsphere,thestandardimmersionoftheproductoftw… 相似文献
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ZHONG ChunPing School of Mathematical Sciences Xiamen University Xiamen China 《中国科学 数学(英文版)》2010,(2)
Let M be a complex n-dimensional manifold endowed with a strongly pseudoconvex complex Finsler metric F. Let M be a complex m-dimensional submanifold of M, which is endowed with the induced complex Finsler metric F. Let D be the complex Rund connection associated with (M, F). We prove that (a) the holomorphic curvature of the induced complex linear connection on (M, F) and the holomorphic curvature of the intrinsic complex Rund connection ~* on (M, F) coincide; (b) the holomorphic curvature of ~* does not exceed the holomorphic curvature of D; (c) (M, F) is totally geodesic in (M, F) if and only if a suitable contraction of the second fundamental form B(·, ·) of (M, F) vanishes, i.e., B(χ, ι) = 0. Our proofs are mainly based on the Gauss, Codazzi and Ricci equations for (M, F). 相似文献
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We investigate rigidity problems for odd-dimensional compact submanifolds.We show that if Mn(n 5)is an odd-dimensional compact submanifold with parallel mean curvature in Sn+p,and if RicM(n-2-1n)(1+H2)and Hδn,whereδn is an explicit positive constant depending only on n,then M is a totally umbilical sphere.Here H is the mean curvature of M.Moreover,we prove that if Mn(n 5)is an odd-dimensional compact submanifold in the space form Fn+p(c)with c 0,and if RicM(n-2-εn)(c+H2),whereεn is an explicit positive constant depending only on n,then M is homeomorphic to a sphere. 相似文献
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We develop the concept of integral Menger curvature for a large class of nonsmooth surfaces. We prove uniform Ahlfors regularity and a C1,λ-a priori bound for surfaces for which this functional is finite. In fact, it turns out that there is an explicit length scale R>0 which depends only on an upper bound E for the integral Menger curvature Mp(Σ) and the integrability exponent p, and not on the surface Σ itself; below that scale, each surface with energy smaller than E looks like a nearly flat disc with the amount of bending controlled by the (local) Mp-energy. Moreover, integral Menger curvature can be defined a priori for surfaces with self-intersections or branch points; we prove that a posteriori all such singularities are excluded for surfaces with finite integral Menger curvature. By means of slicing and iterative arguments we bootstrap the Hölder exponent λ up to the optimal one, λ=1−(8/p), thus establishing a new geometric ‘Morrey–Sobolev’ imbedding theorem.As two of the various possible variational applications we prove the existence of surfaces in given isotopy classes minimizing integral Menger curvature with a uniform bound on area, and of area minimizing surfaces subjected to a uniform bound on integral Menger curvature. 相似文献
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Given a closed connected manifold smoothly immersed in a complete noncompact Riemannian manifold with nonnegative sectional curvature, we estimate the intrinsic diameter of the submanifold in terms of its mean curvature field integral. On the other hand, for a compact convex surface with boundary smoothly immersed in a complete noncompact Riemannian manifold with nonnegative sectional curvature, we can estimate its intrinsic diameter in terms of its mean curvature field integral and the length of its boundary. These results are supplements of previous work of Topping, Wu-Zheng and Paeng. 相似文献
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John Douglas Moore 《Transactions of the American Mathematical Society》2002,354(9):3815-3834
This article develops methods for studying the topology of submanifolds of constant positive curvature in Euclidean space. It proves that if is an -dimensional compact connected Riemannian submanifold of constant positive curvature in , then must be simply connected. It also gives a conformal version of this theorem.
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We give an estimate of the mean curvature of a complete submanifold lying inside a closed cylinder in a product Riemannian manifold . It follows that a complete hypersurface of given constant mean curvature lying inside a closed circular cylinder in Euclidean
space cannot be proper if the circular base is of sufficiently small radius. In particular, any possible counterexample to
a conjecture of Calabi on complete minimal hypersurfaces cannot be proper. As another application of our method, we derive
a result about the stochastic incompleteness of submanifolds with sufficiently small mean curvature.
Dedicated to Professor Manfredo P. do Carmo on the occasion of his 80th birthday. 相似文献
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Ulrich Dierkes Dirk Schwab 《Calculus of Variations and Partial Differential Equations》2005,22(2):173-184
We construct quadratic forms on
which are subharmonic on any n-dimensional minimal submanifold in
and, more generally, on submanifolds of bounded mean curvature. This leads to nonexistence results for connected n-dimensional minimal submanifolds in
as well as to necessary conditions for the existence of connected submanifolds of bounded mean curvature with arbitrary codimension. Furthermore we discuss a barrier principle for n-dimensional submanifolds in
of bounded mean curvature.Received: 11 November 2003, Accepted: 29 January 2004, Published online: 2 April 2004Mathematics Subject Classification (2000):
35J60, 49Q05, 53C42 相似文献
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S. I. Okrut 《Journal of Mathematical Sciences》1991,53(5):526-532
We consider strongly parabolic, Hilbert submanifolds in Riemannian Hilbert manifolds. We prove that their properties are analogous to the known properties in the finite-dimensional case. The main geometric result consists of Theorem 3: a complete, Riemannian, Hilbert submanifold of nonpositive extrinsic curvature and finite codimension in a Hilbert sphere is a great sphere.Translated from Ukrainskii Geometricheskii Sbornik, No. 33, pp. 91–100, 1990.The author expresses his thanks to Professor A. A. Borisenko for posing the problem and guiding the work. 相似文献
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