We derive local integral and sup- estimates for the curvature of stable marginally outer trapped surfaces in a sliced space-time. The estimates bound the shear of a marginally outer trapped surface in terms of the intrinsic and extrinsic curvature of a slice containing the surface. These estimates are well adapted to situations of physical interest, such as dynamical horizons.

We construct noncompact solutions to the affine normal flow of hypersurfaces, and show that all ancient solutions must be either ellipsoids (shrinking solitons) or paraboloids (translating solitons). We also provide a new proof of the existence of a hyperbolic affine sphere asymptotic to the boundary of a convex cone containing no lines, which is originally due to Cheng-Yau. The main techniques are local second-derivative estimates for a parabolic Monge-Amp`ere equation modeled on those of Ben Andrews and Guti´errez-Huang, a decay estimate for the cubic form under the affine normal flow due to Ben Andrews, and a hypersurface barrier due to Calabi.

For all open Riemann surface N and real number  2 (0, /2), we construct a conformal minimal immersion X = (X1,X2,X3) : N ! R3 such that X3+tan()|X1| : N ! R is positive and proper. Furthermore, X can be chosen with an arbitrarily prescribed flux map. Moreover, we produce properly immersed hyperbolic minimal surfaces with non-empty boundary in R3 lying above a negative sublinear graph.

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In this paper, we prove that, a compact complex manifold $X$ admits a smooth Hermitian metric with positive (resp., negative) scalar curvature if and only if $K_X$ (resp., $K_X^{-1}$) is not pseudo-effective. On the contrary, we also show that on an arbitrary compact complex manifold $X$ with complex dimension $\geq 2$, there exist smooth Hermitian metrics with positive total scalar curvature, and one of the key ingredients in the proof relies on a recent solution to the Gauduchon conjecture by G. Székelyhidi, V. Tosatti, and B. Weinkove.