In this paper we survey certain aspects of the classical Weil-Petersson metric and its generalizations. Being a natural L^2 metric on the parameter space of a family of complex manifolds or holomorphic vector bundles which admit some canonical metrics, the Weil-Petersson metric is well defined when the automorphism group of each fiber is discrete and the curvature of the Weil-Petersson metric can be computed via certain integrals over each fiber. We will discuss the case when these fibers have continuous automorphism groups. We also discuss the relation between the Weil-Petersson metric and energy of harmonic maps.

In this paper, we study the Pontryagin numbers of 24 dimensional String manifolds. In partic- ular, we find representatives of an integral basis of the String cobrodism group at dimension 24, based on the work of Mahowald and Hopkins (The structure of 24 dimensional manifolds having normal bundles which lift to B O [8], from “Recent progress in homotopy theory” (D. M. Davis, J. Morava, G. Nishida, W. S. Wilson, N. Yagita, editors), Contemp. Math. 293, Amer. Math. Soc., Providence, RI, 89-110, 2002), Borel and Hirzebruch (Am J Math 80: 459–538, 1958) and Wall (Ann Math 75:163–198, 1962). This has immediate applications on the divisibility of various characteristic numbers of the manifolds. In particular, we establish the 2-primary divisibilities of the signature and of the modified signature coupling with the integral Wu class of Hopkins and Singer (J Differ Geom 70:329–452, 2005), and also the 3- primary divisibility of the twisted signature. Our results provide potential clues to understand a question of Teichner.

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By using certain idea developed in minimal submanifold theory we study rigidity problem for self-shrinkers in the present paper. We prove rigidity results for the squared norm of the second fundamental form of self-shrinkers, either under point-wise conditions or under integral conditions.

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