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Motivated by results on the simplicial volume of locally symmetric spaces of finite volume, in this note, we observe that the simplicial volume of the moduli space Mg,n is equal to 0 if g  2; g = 1, n  3; or g = 0, n  6; and the orbifold simplicial volume of Mg,n is positive if g = 1, n = 0, 1; g = 0, n = 4. We also observe that the simplicial volume of the Deligne-Mumford compactification of Mg,n is equal to 0, and the simplicial volumes of the reductive Borel-Serre compactification of arithmetic locally symmetric spaces ..\X and the Baily-Borel compactification of Hermitian arithmetic locally symmetric spaces ..\X are also equal to 0 if the Q-rank of ..\X is at least 3 or if ..\X is irreducible and of Q-rank 2.

We prove the nonexistence of stable immersed minimal surfaces uniformly conformally equivalent to $ \mathbb{C}$ in any complete orientable four-dimensional Riemannian manifold with uniformly positive isotropic curvature. We also generalize the same nonexistence result to higher dimensions provided that the ambient manifold has uniformly positive complex sectional curvature. The proof consists of two parts: assuming an ``eigenvalue condition'' on the $ \overline {\partial }$-operator of a holomorphic bundle, we prove (1) a vanishing theorem for these holomorphic bundles on $ \mathbb{C}$ and (2) an existence theorem for holomorphic sections with controlled growth by Hörmander's weighted $L^2$-method.

In this paper, we prove the Lipschitz regularity of continuous harmonic maps from an finite dimensional Alexandrov space to a compact smooth Riemannian manifold. This solves a conjecture of F. H. Lin in [36]. The proof extends the argument of Huang-Wang [26].

Entropy is a natural geometric quantity measuring the complexity of a surface embedded in $\mathbb{R}^3$. For dynamical reasons relating to mean curvature flow, Colding-Ilmanen-Minicozzi-White conjectured that the entropy of any closed surface is at least that of the self-shrinking two-sphere. We prove this conjecture for all closed embedded $2$-spheres. Assuming a conjectural Morse index bound (announced recently by Marques-Neves), we can improve the result to apply to all closed embedded surfaces that are not tori. Our results can be thought of as the parabolic analog to the Willmore conjecture and our argument is analogous in many ways to that of Marques-Neves on the Willmore problem. The main tool is the min-max theory applied to the Gaussian area functional in $\mathbb{R}^3$ which we also establish. To any closed surface in $\mathbb{R}^3$ we associate a four parameter canonical family of surfaces and run a min-max procedure. The key step is ruling out the min-max sequence approaching a self-shrinking plane, and we accomplish this with a degree argument. To establish the min-max theory for $\mathbb{R}^3$ with Gaussian weight, the crucial ingredient is a tightening map that decreases the mass of non-stationary varifolds (with respect to the Gaussian metric of $\mathbb{R}^3$) in a continuous manner.