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is a so-called improper integral owing to the infinity in the integrand at x = u. When n = 0 we have associated with (1) the well-known Cauchy principal value, namely (2). Hadamard (1, p. 117 et seq.) derives from an improper integral an expression which he calls its finite part and which, as he shows, possesses many important properties.

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An integral formula and its applications on sub-static manifolds, (with Chao Xia). To appear in Journal of Differential Geometry. A fully-nonlinear flow and quermassintegral inequalities, (with Pengfei Guan). SCIENTIA SINICA Mathematica, Vol. 48, Iss. 1, 2018. An integral formula for affine connections, (with Chao Xia).

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Thesis Title: The Atiyah-Singer Index Formula for Subelliptic Operators on Contact Manifolds Degree Year: 2005 Career Bio: 2006, Lecturer, Mathematics, University of Pennsylvania 2012-, Assistant Professor, Dartmouth College

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Furthermore, I will present the anisotropic counterpart of Allardcompactness theorem for integral varifolds. To conclude, I will focus on the anisotropic isoperimetric problem: I will provide the anisotropic counterpart of Alexandrovcharacterization of volume-constrained critical points among finite perimeter sets.

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