arXiv research

Survey Bernstein-type theorems for graphical surfaces in Euclidean and Lorentz-Minkowski spaces.

The paper generalizes Bour's theorem to Minkowski 4-space.

For entire spacelike stationary 2-dimensional graphs in Minkowski spaces, we establish Bernstein type theorems under specific boundedness assumptions either on the W-function or on the total (Gaussian) curvature. These conclusions imply the classical Bernstein theorem for minimal surfaces in 3-dimensional Euclidean spa…

Classical H.Minkowski theorems on existence and uniqueness of convex polyhedra with prescribed directions and areas of faces as well as the well-known generalization of H.Minkowski uniqueness theorem due to A.D.Alexandrov are extended to a class of nonconvex polyhedra which are called polyhedral herissons and may be de…

Paper generalizes complex Brunn-Minkowski theory and proves new extension theorems.

The paper proves fundamental theorems for timelike surfaces in Minkowski 4-space.

In this paper, we generalize a theorem {à} la Alexandrov of Wang, Wang and Zhang [WWZ] for closed codimension-two spacelike submanifolds in the Minkowski spacetime for an adapted CMC condition .

In this article, we estimate the quasi-local energy with reference to the Minkowski spacetime [16,17], the anti-de Sitter spacetime [4], or the Schwarzschild spacetime [3]. In each case, the reference spacetime admits a conformal Killing-Yano 2-form which facilitates the application of the Minkowski formula in [15] to …

Simplifies Wulff theorem for crystalline shapes using Minkowski Theory.

In this paper, we first establish an equivalence theorem of Minkowski spaces by using results in centro-affine differential geometry. As an application in Finsler geometry, we gives some new characterizations of Berwald spaces.

Paper solves Christoffel-Minkowski and Weingarten curvature problems in hyperbolic space.

Study Minkowski formula for conformal Killing-Yano 2-forms in constant curvature spacetimes.

We obtain isometric minimal helicoidal and rotational surfaces using generalized Bour's theorem in three dimensional Minkowski space. In addition, we show that the surfaces preserve minimality when their Gauss maps identically equal, choosing any diffentiable functions on the profile curve.

The paper derives Gauss-Bonnet theorems for deformed connections in affine and rigid motions groups.

In this paper, we compute sub-Riemannian limits of Gaussian curvature for a Euclidean $C^2$-smooth surface in the affine group and the group of rigid motions of the Minkowski plane away from characteristic points and signed geodesic curvature for Euclidean $C^2$-smooth curves on surfaces. We get Gauss-Bonnet theorems i…

We discuss some aspects of the differential geometry of curves in Minkowski space. We establish the Serret-Frenet equations in Minkowski space and use them to give a very simple proof of the fundamental theorem of curves in Minkowski space. We also state and prove two other theorems which represent Minkowskian versions…

Study anisotropic capillary surfaces in a wedge using generalized Minkowski norms.

The paper proves conditions for zero Gaussian curvature convex hypersurfaces to be hyperplanes.

In this paper, we firstly introduce the group of similarity transformations in the Minkowski-3 space. We describe differential- geometric invariants of a non-lightlike curve according to the group of similarity transformations of the Minkowski 3-space. We show extension of fundamental theorem for non-lightlike curves u…

Study complete 3D λ-translators in Minkowski space with constant properties.

Maximal surfaces in Lorentz-Minkowski space have conjugate graphs.

Paper solves curvature equations in Minkowski space for non-convex domains.

Paper studies singularities of timelike minimal surfaces in Minkowski 3-space.

Unified estimates for mean curvature in Lorentz-Minkowski space.

Calabi's Bernstein-type theorem asserts that a zero mean curvature entire graph in Lorentz-Minkowski space $\boldsymbol L^3$ which admits only space-like points is a space-like plane. Using the fluid mechanical duality between minimal surfaces in Euclidean 3-space $\boldsymbol E^3$ and maximal surfaces in Lorentz-Minko…

We prove that an approximated version of the Brunn--Minkowski inequality with volume distortion coefficient implies a Gaussian concentration-of-measure phenomenon. Our main theorem is applicable to discrete spaces.

Researchers prove uniqueness and continuity of solution to L_p dual Minkowski problem.

The paper proves a Minkowski-like theorem for tetrahedra in dS3 and AdS3.

In this paper, we investigate the similarity transformations in the Minkowski-n space. We study the geometric invariants of non-null curves under the similarity transformations. Besides, we extend the fundamental theorem for a non-null curve according to a similarity motion. We determine all non-null self-similar curve…

This is a survey of results on positivity of vector bundles, inspired by the Brunn-Minkowski and Prékopa theorems. Applications to complex analysis, Kähler geometry and algebraic geometry are also discussed.

In this paper, we introduce isoparametric functions and isoparametric hypersurfaces in Finsler manifolds and give the necessary and sufficient conditions for a transnormal function to be isoparametric. We then prove that hyperplanes, Minkowski hyperspheres and $F^*$-Minkowski cylinders in a Minkowski space with $BH$-vo…

Paper proves rigidity results for anisotropic capillary hypersurfaces.

The paper examines isometric timelike surfaces in 4D Minkowski space.

The paper discusses rigidity results for inequalities on weighted Riemannian manifolds.

Sharp inequality proved in 3D hyperbolic spaces using flow methods.

We establish a Positive Mass Theorem for initial data sets of the Einstein equations having generalized trapped surface boundary. In particular we answer a question posed by R. Wald concerning the existence of generalized apparent horizons in Minkowski space.

A mechanical linkage is a mechanism made of rigid rods linked together by flexible joints, in which some vertices are fixed and others may move. The partial configuration space of a linkage is the set of all the possible positions of a subset of the vertices. We characterize the possible partial configuration spaces of…

We obtain a gradient estimate for the Gauss maps from complete spacelike constant mean curvature hypersurfaces in Minkowski space into the hyperbolic space. As applications, we prove a Bernstein theorem which says that if the image of the Gauss map is bounded from one side, then the spacelike constant mean curvature hy…

Study complete space-like stationary surfaces with graphical Gauss image, estimating exceptional values and classifying degenerate surfaces.

The paper studies helicoidal surfaces of non-lightlike frontals in Lorentz-Minkowski 3-space.

A marginally trapped surface in the four-dimensional Minkowski space is a spacelike surface whose mean curvature vector is lightlike at each point. We associate a geometrically determined moving frame field to such a surface and using the derivative formulas for this frame field we obtain seven invariant functions. Our…

The paper proves approximation and interpolation theorems for maxfaces with singularities.

The paper extends Bernstein Theorem for minimal spacelike surfaces in 4D Minkowski space.

Calabi and Cheng-Yau's Bernstein-type theorem asserts that an entire zero mean curvature graph in Lorentz-Minkowski $(n+1)$-space $\boldsymbol R^{n+1}_1$ which admits only space-like points is a hyperplane. Recently, the third and fourth authors proved a line theorem for hypersurfaces at their degenerate light-like poi…

In this paper, by the studying of the Gauss map, Laplacian operator, curvatures of surfaces in $\mathbb{R}_{1}^{3}$ and Bour's theorem, we are going to identify surfaces of revolution with pointwise 1-type Gauss map property in $3-$dimensional Minkowski space.

Classification theorems for Ricci solitons on Minkowski hypersurfaces.

The paper quantifies how much of a 4-ball must be removed to squeeze into a cylinder, proving a lower bound on the Minkowski dimension.

Study timelike surfaces with parallel mean curvature in Minkowski 4-space.