Contemporary Geometry: J.-Q. Zhong Memorial Volume by Hung-Hsi Wu (auth.), Hung-Hsi Wu (eds.)

By Hung-Hsi Wu (auth.), Hung-Hsi Wu (eds.)

Early one morning in April of 1987, the chinese language mathematician J. -Q. Zhong died suddenly of a middle assault in long island. He was once then close to the top of a one-year stopover at within the usa. whilst information of his dying reached his Chinese-American acquaintances, it was once instantly determined by means of all and sundry that anything might be performed to maintain his reminiscence. the current quantity is an outgrowth of this sentiment. His buddies in China have additionally demonstrated a Zhong Jia-Qing Memorial Fund, which has given that two times presented the Zhong Jia-Qing prizes for chinese language arithmetic graduate scholars. it truly is was hoping that no less than a part of the explanations for the esteem and affection within which he was once held by means of all who knew him may come via within the succeeding pages of this quantity. the 3 survey chapters through Li and Treibergs, Lu, and Siu (Chapters 1-3) all focus on the parts of arithmetic within which Zhong made noteworthy contributions. as well as placing Zhong's mathematical contributions in standpoint, those articles can be priceless additionally to a wide phase of the mathematical neighborhood; jointly they provide a coherent photograph of a large component of modern geometry. The survey of Lu differs from the opposite in that it offers a firsthand account of the paintings performed within the People's Republic of China in different advanced variables within the final 4 decades.

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Suppose that M" is a compact manifold whose Ricci curvature satisfies Rij ~ (n -1)k8ij for some constant k> O. Suppose that AI(M) = AI(S") = kn, where S" is the standard round sphere of constant sectional curvature k. Then M is isometric to S". There are corresponding statements for manifolds with boundary. For the Dirichlet problem the result was achieved by Reilly [74], for the Neumann problem by Escobar [37]. 3. Suppose M" is a compact manifold with boundary such that the Ricci curvature satisfies Rij ~ (n - 1) k8ij for some constant k > O.

Tl(N)1 is the order of the fundamental group. This area estimate is then utilized by Choi and Schoen [29] to obtain the following important consequence about the compactness of the space of minimal surfaces. 35. Let N 3 be a compact manifold with positive Ricci curvature. Suppose M is an embedded compact minimal surface in N. Then there is a constant C depending only on N and the Euler characteristic ofM such that sup Ihij I :5 C, M where hij is the second fundamental form of M. Hence, the space of compact, minimal, embedded surfaces MeN of a fixed topological type is compact in the cj topology for any j ~ 2.

Inst. Fourier 30, 109-128 (1980). 9. G. Besson, On the multiplicity of the eigenvalues of the Laplacian, in Geometry and Analysis on Manifolds, (T. , Vol. 32-53, Springer-Verlag, New York (1988). 10. S. Bochner, Vector fields and Ricci curvature, Bull. Am. Math. Soc. 52, 776-797 (1946). Eigenvalue Techniques in Geometry 49 11. E. Bombieri, Theory of minimal surfaces and a counterexample to the Bernstein conjecture in high dimensions, Lecture notes. 12. R. Brooks, The fundamental group and the spectrum of the Laplacian, Comment.

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