Related papers: Calabi affine maximal surfaces and centroaffine Be…

Complex Ricci-flat (i.e., Calabi-Yau) hypersurfaces in spaces admitting a maximal (toric) $U(1)^n$ gauge symmetry of general type (encoded by certain non-convex and multi-layered multitopes) may degenerate, but can be smoothed by rational…

We introduce a new family of affine metrics on a locally strictly convex surface $M$ in affine 4-space. Then, we define the symmetric and antisymmetric equiaffine planes associated with each metric. We show that if $M$ is immersed in a…

The first part of this paper considers higher order CR invariants of three dimensional hypersurfaces of finite type. Using a full normal form we give a complete characterization of hypersurfaces with trivial local automorphism group, and…

It is well known that the Fano scheme of lines on a cubic 4-fold is a symplectic variety. We generalize this fact by constructing a closed p-form with p=2n-4 on the Fano scheme of lines on a (2n-2)-dimensional hypersurface Y of degree n. We…

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

We establish the convergence theory of multiplicative Diophantine approximation for all non-degenerate, smooth manifolds. We also settle said convergence theory for all affine subspaces satisfying a highly generic and essentially optimal…

In this paper, we study locally strongly convex Tchebychev hypersurfaces, namely the {\it centroaffine totally umbilical hypersurfaces}, in the $(n+1)$-dimensional affine space $\mathbb{R}^{n+1}$. We first make an ordinary-looking…

We use Cartan's method of moving frames to compute a complete set of local invariants for nondegenerate, 2-dimensional centroaffine surfaces in $\mathbb{R}^5 \setminus \{0\}$ with nondegenerate centroaffine metric. We then give a complete…

This paper explores equi-centro-affine extremal hypersurfaces in an ellipsoid. By analyzing the evolution of invariant submanifold flows under centro-affine unimodular transformations, we derive the first and second variational formulas for…

We prove some Bernstein theorems for entire space-like submanifolds in pseudo-Euclidean spaces and, as a corollary, we obtain a new proof of the Calabi-Pogorelov theorem on global solutions of Monge-Ampere equations.

In the setting of fractional minimal surfaces, we prove that if two nonlocal minimal sets are one included in the other and share a common boundary point, then they must necessarily coincide. This strict maximum principle is not obvious,…

In this paper we establish some parabolicity criteria for maximal surfaces immersed into a Lorentzian product space of the form $M^2\times\mathbb{R}_1$, where $M^2$ is a connected Riemannian surface with non-negative Gaussian curvature and…

We generalize a Bernstein-type result due to Albujer and Al\'ias, for maximal surfaces in a curved Lorentzian product 3-manifold of the form $\Sigma_1\times \mathbb{R}$, to higher dimension and codimension. We consider $M$ a complete…

Following the work of Castano-Bernard and Matessi on conifold transition in the Gross-Siebert program, we construct orbi-conifold transitions of the Shoen's Calabi-Yau threefold and their mirrors. The construction glues together the local…

Using the fractional discrete Laplace operator for triangle meshes, we introduce a fractional combinatorial Calabi flow for discrete conformal structures on surfaces, which unifies and generalizes Chow-Luo's combinatorial Ricci flow for…

We apply recently developed convex programs to find the minimal-area Riemannian metric on $2n$-sided polygons ($n\geq 3$) with length conditions on curves joining opposite sides. We argue that the Riemannian extremal metric coincides with…

This is the sequel to arXiv:math/0001089. In this paper, we complete the promised description of moduli of abelian surfaces of low degree, covering the cases of degree (1,12), (1,14), (1,16), (1,18) and (1,20). In each case, we describe…

We construct normal forms for Levi degenerate hypersurfaces of finite type in $\mathbb C^2$. As one consequence, an explicit solution to the problem of local biholomorphic equivalence is obtained. Another consequence determines the…

We continue to develop our method for effectively computating the special K\"ahler geometry on the moduli space of Calabi-Yau manifolds. We generalize it to all polynomial deformations of Fermat hypersurfaces.

By introducing a more flexible notion of convexity, we obtain a new Omori-Yau maximum principle for harmonic maps. In the spirit of the Calabi-Yau conjectures, this principle is more suitable for studying the unboundedness of certain…