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The first eigenvalue of Laplace operator under powers of mean curvature flow

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Science china Mathematics 2010年第7期53卷 1702-1709页

作者： ZHAO Liang center of mathematical sciences, zhejiang university, hangzhou 31002., china 1. center of mathematical sciences Zhejiang University Hangzhou 310027 China

In this paper, we derive the evolution equation for the first eigenvalue of Laplace operator along powers of mean curvature flow. Considering a compact, strictly convex n-dimensional surface M without boundary, which is smoothly immersed in R n+1 , we prove that if the initial 2-dimensional surface M is totally umbilical, then the first eigenvalue is nondecreasing along the unnormalized H k -flow. Moreover, as applications of the evolution equation, we construct some monotonic quantities along this kind of flow.

关键词： Laplace operator H k -flow eigenvalue monotonicity

TRIPLE　INTERACTIONS　OF　CONORMAL　WAVES　FOR　HIGHER　ORDER　SEMILINEAR　HYPERBOLIC　EQUMIONS

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Chinese Annals of Mathematics,Series B 1995年第1期16卷 23-32页

作者： FANG DAOYUAN (center for mathematical sciences, zhejiang university, hangzhou 31002., china.) 浙江大学数学科学中心浙江杭州

The interaction of three conormal waves for semi-linear strictly hyperbolic equations of third order is considered. Let Σi, i = 1, 2, 3, be smooth characteristic surfaces for P= Da(D_t￣2 -△) intersecting transversally at the origin. Suppose that the solution u to Pu = f(t, x ,y)D u), ≤2 is conormal to Σi, i = 1, 2, 3, for t 3.

关键词： Hyperbolic equation Conormal wave Interaction Characteristic surface.

Asymptotic Behavior of Gibbs Functions for M-Band Wavelet Expansions

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Acta Mathematica Sinica,English Series 1999年第2期15卷 165-172页

作者： Daren Huang Zeyin Zhang, center for mathematical sciences, zhejiang university, hangzhou 31002., P. R. china center for mathematical sciences Zhejiang University Hangzhou P. R. China

In this paper the asymptotic behavior of Gibbs function for a class of M-band wavelet expansions is given. In particular, the Daubechies’ wavelets are included in this class.

关键词： Gibbs phenomenon Wavelet expansion Filter Scaling function Vanishing moment

Complete hypersurfaces in a 4-dimensional hyperbolic space

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Applied Mathematics(A Journal of Chinese Universities) 2009年第3期24卷 370-378页

作者： XU Hong-wei1 ZHAO En-tao2 center of mathematical sciences,zhejiang university,hangzhou 31002.,china center of mathematical sciences Zhejiang University Hangzhou 310027 China

This paper gives a classification of complete hypersurfaces with nonzero constant mean curvature and constant quasi-Gauss-Kronecker curvature in the hyperbolic space H4（-1）,whose scalar curvature is bounded from below.

关键词： complete hypersurface mean curvature quasi-Gauss-Kronecker curvature

On a Class of Finitely Supported Scaling Functions with Properties of Interpolation and Symmetry

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Acta Mathematica Sinica,English Series 2001年第2期17卷 333-338页

作者： Ze Yin ZHANG center for mathematical sciences. zhejiang university. hangzhou 31002.. P. R. china center for mathematical sciences Zhejiang University Hangzhou 310027 P. R. China

In this paper, a class of finitely supported orthogonal filters with interpolation and sym metry is constructed.

关键词： Filter Scaling function

Time-periodic solutions of the Einstein's field equations II:geometric singularities

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Science china Mathematics 2010年第6期53卷 1507-1520页

作者： KONG DeXing1,LIU KeFeng2,& SHEN Ming3 1Department of Mathematics,zhejiang university,hangzhou 31002.,china 2Department of Mathematics,university of California at Los Angeles,CA 90095,USA 3center of mathematical sciences,zhejiang university,hangzhou 31002.,china 1. Department of Mathematics Zhejiang University Hangzhou 310027 China2. Department of Mathematics University of California at Los Angeles Los Angeles CA 90095 USA3. Center of Mathematical Sciences Zhejiang University Hangzhou 310027 China

In this paper,we construct several kinds of new time-periodic solutions of the vacuum Einstein's field equations whose Riemann curvature tensors vanish,keep finite or take the infinity at some points in these space-times,*** singularities of these new time-periodic solutions are investigated and some new physical phenomena are discovered.

关键词： Einstein’s field equations time-periodic solution Riemann curvature tensor singularity event horizon

Lipschitz continuity of best approximations and Chebyshev centers

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Chinese Science Bulletin 1998年第3期43卷 185-188页

作者： LI Chong 1 and WANG Xinghua 2 1. center for mathematical sciences, zhejiang university, hangzhou 31002., china 2. Department of Mathematics, hangzhou university, hangzhou 31002., china center for mathematical sciences Zhejiang University Hangzhou China Department of Mathematics Hangzhou University Hangzhou China

The Lipschitz continuity results are given for best approximations and Chebyshev centers. Consequently, the open problem presented by Septycki and Vleck is solved completely.

关键词： best approximation Chebyshev center uniform convexity.

SOBOLEV INEQUALITY ON RIEMANNIAN MANIFOLDS

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Chinese Annals of Mathematics,Series B 2005年第4期26卷 651-658页

作者： WANG MENG Department of Mathematics, zhejiang university, hangzhou 31002., china School of mathematical sciences, Pudan university, Shanghai 200433, china. Department of Mathematics Zhejiang University Hangzhou 310027 China. School of mathematical sciences Fudan University Shanghai 200433 China

Let M be an n dimensional complete Riemannian manifold satisfying the doublingvolume property and an on-diagonal heat kernel estimate. The necessary-sufficientcondition for the Sobolev inequality ‖f‖q ≤ Cn,,v,p,q(... 详细信息

关键词： Sobolev inequality Complete manifold Riesz transform Potential Heat kernel

ON COMPLETE SUBMANIFOLDS WITH PARALLEL MEAN CURVATURE IN NEGATIVE PINCHED MANIFOLDS

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Applied Mathematics(A Journal of Chinese Universities) 2007年第2期22卷 153-162页

作者： Leng Yan Xu Hongwei zhejiang university, center of mathematical sciences Eangzhou 31002., china zhejiang university, center of mathematical sciences Eangzhou 31002., china zhejiang university Center of Mathematical Sciences hangzhou 31002. China

A rigidity theorem for oriented complete submanifolds with parallel mean curvature in a complete and simply connected Riemannian （n ＋ p）-dimensional manifold N^n＋p with negative sectional curvature is proved. For given positive integers n（≥ 2）, p and for a constant H satisfying H 〉 1 there exists a negative number τ（n,p, H） ∈ （-1, 0） with the property that if the sectional curvature of N is pinched in [-1, τ-（n,p, H）], and if the squared length of the second fundamental form is in a certain interval, then N^n＋p is isometric to the hyperbolic space H^n＋P（-1）. As a consequence, this submanifold M is congruent to S^n（1√H^2 - 1） or the Veronese surface in S^4（1/√H^2-1）.

关键词： complete submanifold, rigidity theorem, mean curvature, second fundamental form pinchedRiemannian manifold