We review some recent results on the mean curvature flows of Lagrangian submanifolds from the perspective of geometric partial differential equations. These include global existence and convergence results, characterizations of first-time singularities, and constructions of self-similar solutions.

By adapting the test functions introduced by Choi-Daskaspoulos \cite{c-d} and Brendle-Choi-Daskaspoulos \cite{b-c-d} and exploring properties of the $k$-th elementary symmetric functions $\sigma_{k}$ intensively, we show that for any fixed $k$ with $1\leq k\leq n-1$, any strictly convex closed hypersurface in $\mathbb{R}^{n+1}$ satisfying $\sigma_{k}^{\alpha}=\langle X,\nu \rangle$, with $\alpha\geq \frac{1}{k}$, must be a round sphere.

If 𝑈:[0,+∞[×𝑀 is a uniformly continuous viscosity solution of the evolution Hamilton-Jacobi equation ∂_𝑡𝑈+𝐻(𝑥,∂_𝑥 𝑈)=0, where 𝑀 is a not necessarily compact manifold, and 𝐻 is a Tonelli Hamiltonian, we prove the set Σ(𝑈), of points in ]0,+∞[×𝑀 where 𝑈 is not differentiable, is locally contractible. Moreover, we study the homotopy type of Σ(𝑈). We also give an application to the singularities of the distance function to a closed subset of a complete Riemannian manifold.

We study the topology of a complete asymptotically hyperbolic Einstein manifold of which its conformal boundary has positive Yamabe invariant. We prove that all maps from such manifold into any nonpositively curved manifold are homotopically trivial. Our proof is based on a Bochner type argument on harmonic maps.

We establish a boundary maximum principle for free boundary minimal submanifolds in a Riemannian manifold with boundary, in any dimension and codimension. Our result holds more generally in the context of varifolds.