Sharp reverse affine isoperimetric inequalities for asymmetric Wulff shapes and their polars are established, along with the characterization of all extremals. These new inequalities have as special cases previously obtained simplex inequalities by Ball, Barthe and Lutwak, Yang, and Zhang. In particular, they provide the solution to a problem by Zhang.

In this paper, we investigate the space of certain weak stability conditions on the triangulated category of D0-D2-D6 bound states on a smooth projective Calabi-Yau 3-fold. In the case of a quintic 3-fold, the resulting space is interpreted as a universal covering space of an infinitesimal neighborhood of the conifold point in the stringy K¨ahler moduli space. We then associate the DT type invariants counting semistable objects, which give new curve counting invariants on Calabi-Yau 3-folds. We also investigate the wall-crossing formula of our invariants and their interplay with the Seidel-Thomas twist.

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In this paper we derive optimal growth estimates on the potential functions of complete noncompact shrinking solitons. Based on this, we prove that a complete noncompact gradient shrinking Ricci soliton has at most Euclidean volume growth. This latter result can be viewed as an analog of the well-known volume comparison theorem of Bishop that a complete noncompact Riemannian manifold with nonnegative Ricci curvature has at most Euclidean volume growth.

A well-known conjecture of Yau states that the first eigenvalue of every closed minimal hypersurface Mn in the unit sphere S^(n+1)(1) is just its dimension n. The present paper shows that Yau conjecture is true for minimal isoparametric hypersurfaces. Moreover, the more fascinating result of this paper is that the first eigenvalues of the focal submanifolds are equal to their dimensions in the non-stable range.