We give a new proof of a classical uniqueness theorem of Alexandrov [4] using the weak uniqueness continuation theorem of BersNirenberg [8]. We prove a version of this theorem with the minimal regularity assumption: the spherical Hessians of the corresponding convex bodies as Radon measures are nonsingular.

The chapter discusses the group actions on R³. The Smith conjecture has many equivalent forms, and each of these forms has various consequences and generalizations. The Smith conjecture is a structure theorem about symmetries of the product of a compact surface with an interval. Here the interval may be closed or open. The usual Smith conjecture is equivalent to proving the smooth <i>Z_n</i> actions on <i>S</i>² [0, 1] are conjugate to actions that preserve the product structure. This generalized Smith conjecture represents the belief that all the symmetries of the product of a compact surface with an interval actually arise from the symmetries of the surface extended trivially to the product structure. The chapter presents how to apply minimal surfaces to study the generalized Smith conjecture on <i>S</i>² <i>I</i>, where <i>I</i> is an open or closed interval. A geodesic version of the loop theorem and a new equivariant

We investigate the Kähler-Ricci flow on holomorphic fiber spaces whose generic fiber is a Calabi-Yau manifold. We establish uniform metric convergence to a metric on the base, away from the singular fibers, and show that the rescaled metrics on the fibers converge to Ricci-flat Kähler metrics. This strengthens previous work of Song-Tian and others. We obtain analogous results for degenerations of Ricci-flat Kähler metrics.

In relativity, the energy of a moving particle depends on the observer, and the rest mass is the minimal energy seen among all observers. The Wang–Yau quasi-local mass for a surface in spacetime introduced in Wang and Yau (Phys Rev Lett 102(2):021101, 2009 ; Commun Math Phys 288(3):919–942, 2009) is defined by minimizing quasi-local energy associated with admissible isometric embeddings of the surface into the Minkowski space. A critical point of the quasi-local energy is an isometric embedding satisfying the Euler–Lagrange equation. In this article, we prove results regarding both local and global minimizing properties of critical points of the Wang–Yau quasi-local energy. In particular, under a condition on the mean curvature vector we show a critical point minimizes the quasi-local energy locally. The same condition also implies that the critical point is globally minimizing among all axially symmetric embedding provided the image of the associated isometric embedding lies in a totally geodesic Euclidean 3-space.

We apply the mean curvature flow to deform symplectomorphisms of $\mathbb{CP}^n$. In particular, we prove that, for each dimension $n$, there exists a constant , explicitly computable, such that anypinched symplectomorphism of $\mathbb{CP}^n$ is symplectically isotopic to a biholomorphic isometry.