We consider an embedded convex compact ancient solution t to the curve shortening flow in R2. We prove that t is either a family of contracting circles, which is a type I ancient solution, or a family of evolving Angenent ovals, which is of type II.

By the integral method, we prove that any space-like entire graphic self-shrinking solution to Lagrangian mean curvature flow in $\mathbb{R}^{2n}_{n}$ with the indefinite metric $\sum_i dx_idy_i$ is flat. This result improves the previous ones in [9] and [1] by removing the additional assumption in their results. In a similar manner, we reprove its Euclidean counterpart which is established in [1].

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.

Let  $\sum$ be a complete minimal Lagrangian submanifold of $\mathbb{C}^n$. We identify regions in the Grassmannian of Lagrangian subspaces so that whenever the image of the Gauss map of $\sum$ lies in one of these regions, then  $\sum$ is an affine space.

Let M 4n be a complete quaternionic K¨ahler manifold with scalar curvature bounded below by .16n(n + 2). We get a sharp estimate for the first eigenvalue λ1(M) of the Laplacian, which is λ1(M) ≤ (2n + 1)2. If the equality holds, then either M has only one end, or M is diffeomorphic to R × N with N given by a compact manifold. Moreover, if M is of bounded curvature, M is covered by the quaterionic hyperbolic space QHn and N is a compact quotient of the generalized Heisenberg group. When λ1(M) ≥ 8(n+2) 3 , we also prove that M must have only one end with infinite volume.