共查询到20条相似文献,搜索用时 15 毫秒
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Koichi Nagano 《Mathematische Zeitschrift》2002,241(1):127-163
We obtain a volume convergence theorem for Alexandrov spaces with curvature bounded above with respect to the Gromov-Hausdorff
distance. As one of the main tools proving this, we construct an almost isometry between Alexandrov spaces with curvature
bounded above, with weak singularities, which are close to each other. Furthermore, as an application of our researches of
convergence phenomena, for given positive integer , we prove that, if a compact, geodesically complete, n-dimensional CAT(1)-space has the volume sufficiently close to that of the unit n-sphere, then it is bi-Lipschitz homeomorphic to the unit n-sphere.
Received: 30 January 2001; in final form: 30 October 2001 / Published online: 4 April 2002 相似文献
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Distance functions of metric spaces with lower curvature bound, by definition, enjoy various metric inequalities; triangle
comparison, quadruple comparison and the inequality of Lang–Schroeder–Sturm. The purpose of this paper is to study the extremal
cases of these inequalities and to prove rigidity results. The spaces which we shall deal with here are Alexandrov spaces
which possibly have infinite dimension and are not supposed to be locally compact. 相似文献
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Ayato Mitsuishi 《Geometriae Dedicata》2010,144(1):101-114
We prove a splitting theorem for Alexandrov space of nonnegative curvature without properness assumption. As a corollary,
we obtain a maximal radius theorem for Alexandrov spaces of curvature bounded from below by 1 without properness assumption.
Also, we provide new examples of infinite dimensional Alexandrov spaces of nonnegative curvature. 相似文献
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Fabrizio Cacciafesta 《Journal of Geometry》1982,18(1):5-8
By a theorem of A. D. Alexandrov a bijection : R
n
n
Rn (n3) satisfiyng d(a,b)=0d ((a),(b))=0 a, b Rn (d distance associated to the Minkowski metric) is a Lorentz transformation up to a dilatation. We prove that the inverse implication in the condition above may be dropped, being a consequence of the direct one.Dedicated to Raphael Artzy on the occasion of his 70th birthday 相似文献
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Leung-Fu Cheung 《manuscripta mathematica》1991,70(1):219-226
Summary In this paper we present sufficient conditions for the non-existence of stable, complete, noncompact, constant mean curvature
hypersurfaces in certain (n+1)-dimensional Riemannian manifolds. 相似文献
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Hong-Wei Xu 《Archiv der Mathematik》1993,61(5):489-496
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Let M be a compact hypersurface with constant mean curvature in Denote by H and S the mean curvature and the squared norm of the second fundamental form of M,respectively.We verify that there exists a positive constant γ(n) depending only on n such that if |H| ≤γ(n) and β(n,H)≤S ≤β(n,H)+n/18,then S≡β(n,H) and M is a Clifford torus.Here,β(n,H)=n+n~3/2(n-1)H~2+n(n-2)/2(n-1)(1/2)n~2H~4+4(n-1)H~2. 相似文献
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Huai-Dong Cao Haizhong Li 《Calculus of Variations and Partial Differential Equations》2013,46(3-4):879-889
In this paper, we prove a classification theorem for self-shrinkers of the mean curvature flow with |A|2 ≤ 1 in arbitrary codimension. In particular, this implies a gap theorem for self-shrinkers in arbitrary codimension. 相似文献
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Huai-Dong Cao Ying Shen Shunhui Zhu 《Calculus of Variations and Partial Differential Equations》1998,7(2):141-157
We obtain a gradient estimate for the Gauss maps from complete spacelike constant mean curvature hypersurfaces in Minkowski
space into the hyperbolic space. As an application, 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 extends
the previous theorems obtained by B. Palmer [Pa] and Y.L. Xin [Xin1] where they assume that the image of the Gauss map is
bounded. We also prove a Bernstein theorem for spacelike complete surfaces with parallel mean curvature vector in four-dimensional
spaces.
Received July 4, 1997 / Accepted October 9, 1997 相似文献
14.
LetM be a compact minimal surface inS
3. Y. J. Hsu[5] proved that if S222, thenM is either the equatorial sphere or the Clifford torus, whereS is the square of the length of the second fundamental form ofM, ·2 denotes theL
2-norm onM. In this paper, we generalize Hsu's result to any compact surfaces inS
3 with constant mean curvature.Supported by NSFH. 相似文献
15.
Let be a compact immersed surface in the unit sphere with constant mean curvature . Denote by the linear map from into , , where is the linear map associated to the second fundamental form and is the identity map. Let denote the square of the length of . We prove that if , then is either totally umbilical or an -torus, where is a constant depending only on the mean curvature .
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We generalize the second pinching theorem for minimal hypersurfaces in a sphere due to Peng–Terng, Wei–Xu, Zhang, and Ding–Xin to the case of hypersurfaces with small constant mean curvature. Let $M^n$ be a compact hypersurface with constant mean curvature $H$ in $S^{n+1}$ . Denote by $S$ the squared norm of the second fundamental form of $M$ . We prove that there exist two positive constants $\gamma (n)$ and $\delta (n)$ depending only on $n$ such that if $|H|\le \gamma (n)$ and $\beta (n,H)\le S\le \beta (n,H)+\delta (n)$ , then $S\equiv \beta (n,H)$ and $M$ is one of the following cases: (i) $S^{k}\Big (\sqrt{\frac{k}{n}}\Big )\times S^{n-k}\Big (\sqrt{\frac{n-k}{n}}\Big )$ , $\,1\le k\le n-1$ ; (ii) $S^{1}\Big (\frac{1}{\sqrt{1+\mu ^2}}\Big )\times S^{n-1}\Big (\frac{\mu }{\sqrt{1+\mu ^2}}\Big )$ . Here $\beta (n,H)=n+\frac{n^3}{2(n-1)}H^2+\frac{n(n-2)}{2(n-1)} \sqrt{n^2H^4+4(n-1)H^2}$ and $\mu =\frac{n|H|+\sqrt{n^2H^2+ 4(n-1)}}{2}$ . 相似文献
17.
Baoqiang Wu 《Proceedings of the American Mathematical Society》2004,132(1):211-215
In this paper we prove a general Bernstein theorem on the complete spacelike constant mean curvature hypersurfaces in Minkowski space. The result generalizes the previous result of Cao-Shen-Zhu (1998) and Xin (1991). The proof again uses the fact that the Gauss map of a constant mean curvature hypersurface is harmonic, which was proved by K. T. Milnor (1983), and the maximum principle of S. T. Yau (1975).
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Vicente Miquel 《Annals of Global Analysis and Geometry》1994,12(1):211-218
We prove that every connected compact Hopf hypersurface of a complex space form, contained in a geodesic ball of radius strictly smaller than the injectivity radius of, having constant mean curvature and with if if < 0 is a geodesic sphere of.Work partially supported by DGICYT Grant No. PB91-0324. 相似文献
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Santiago R. Simanca 《Israel Journal of Mathematics》1992,78(1):85-93
By working in ? n with potentials of the forma logu + s(u), u the square of the distant to the origin, we obtain extremal Kähler metrics of nonconstant scalar curvature on the blow-up of ? n at \(\vec 0\) . We then show that these metrics can be completed at ∞ by adding a ?? n?1, and reobtain the extremal Kähler metrics of non-constant scalar curvature constructed by Calabi on the blow-up of ?? n at one point. A similar construction produces this type of metrics on other bundles over ?? n ? 1. 相似文献
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The McKay conjecture asserts that for every finite group G and every prime p, the number of irreducible characters of G having p’-degree is equal to the number of such characters of the normalizer of a Sylow p-subgroup of G. Although this has been confirmed for large numbers of groups, including, for example, all solvable groups and all symmetric
groups, no general proof has yet been found. In this paper, we reduce the McKay conjecture to a question about simple groups.
We give a list of conditions that we hope all simple groups will satisfy, and we show that the McKay conjecture will hold
for a finite group G if every simple group involved in G satisfies these conditions. Also, we establish that our conditions are satisfied for the simple groups PSL2(q) for all prime powers q≥4, and for the Suzuki groups Sz(q) and Ree groups R(q), where q=2
e
or q=3
e
respectively, and e>1 is odd. Since our conditions are also satisfied by the sporadic simple group J
1, it follows that the McKay conjecture holds (for all primes p) for every finite group having an abelian Sylow 2-subgroup. 相似文献