arXiv research

Minimal normal curvature immersions in the unit ball studied.

Veronese minimizes normal curvatures to sphere.

We establish a nice orthonormal frame field on a closed surface minimally immersed in a unit sphere $S^{n}$, under which the shape operators take very simple forms. Using this frame field, we obtain an interesting property $K+K^{N}=1$ for the Gauss curvature $K$ and the normal curvature $K^{N}$ if the Gauss curvature i…

In this paper we consider Lorentzian surfaces in the 4-dimensional pseudo-Riemannian sphere $\mathbb S^4_2(1)$ with index 2 of curvature one. We obtain the complete classification of minimal Lorentzian surfaces $\mathbb S^4_2(1)$ whose Gaussian and normal curvatures are constants. We conclude that such surfaces have th…

Paper proves properties of minimal hypersurfaces in specific solitons.

We consider the isometric deformation problem for oriented non simply connected immersed minimal surfaces $f:M \to S^{4}$. We prove that the space of all isometric minimal immersions of $M$ into $S^{4}$ with the same normal curvature function is, within congruences, either finite or a circle. Furthermore, we show that …

Lower bounds on average normal curvature for submanifolds in Riemannian domains.

In this paper, we proved the Normal Scalar Curvature Conjecture and the Bottcher-Wenzel Conjecture. We also established some new pinching theorems for minimal submanifolds in spheres.

Sharp curvature bounds for minimal graphs over unit disk.

The paper finds a Weierstrass representation for a specific type of Lorentzian minimal surface.

We consider graphs Sigma^n in R^m with prescribed mean curvature and flat normal bundle. Using techniques of Schoen, Simon and Yau, and Ecker-Huisken, we derive an interior curvature estimate of the form |A|^2<=C/R^2 up to dimension n<=5, where C is a constant depending on natural geometric data of Sigma^n only. This g…

In this overview report we generalize Erhard Heinz' curvature estimate for minimal graphs in R^3 to graphs in R^n of prescribed mean curvature. Secondly, we analyse these problems in the frame of the outer differential geometry which leads us to the notions of normal torsion and normal curvature for immersions in R^4.

We study minimal Lorentz surfaces in the pseudo-Euclidean 4-space with neutral metric whose first normal space is two-dimensional and whose Gauss curvature $K$ and normal curvature $\varkappa$ satisfy the inequality $K^2-\varkappa^2 >0$. Such surfaces we call minimal Lorentz surfaces of general type. On any surface of …

The Kähler-Ricci flow yields bounded diameter and Ricci curvature for minimal models.

Defines and analyzes generalized normal ruled surfaces of curves in 3D space.

The study pinches rigidity theorems for minimal submanifolds in spheres.

It is well-known that a minimal graph of codimension one is stable, i.e. the second variation of the area functional is non-negative. This is no longer true for higher codimensional minimal graphs. In this note, we prove that a minimal graph of any codimension is stable if its normal bundle is flat. We also prove minim…

The study examines minimal surfaces in Riemannian products of surfaces.

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

Study of timelike surfaces in Minkowski space with specific geometric properties.

Lu conjecture proven for minimal 2-spheres and surfaces under certain conditions.

AQFC method estimates mesh curvatures using quadratic surfaces.

The aim of this paper is to investigate the differential geometry of immersed surfaces in three-dimensional normed spaces from the viewpoint of affine differential geometry. We endow the surface with a useful Riemannian metric which is closely related to normal curvature, and from this we re-calculate the Minkowski Gau…

The following results are proved: Theorem 1. A totally real semiparallel submanifold of constant curvature with parallel f-structure in the normal bundle of a Kähler manifold N is flat or a totally geodesic submanifold of N. Theorem 2. A totally real minimal semiparallel submanifold M with parallel f-structure in the n…

A space-like surface in Minkowski space-time is minimal if its mean curvature vector field is zero. Any minimal space-like surface of general type admits special isothermal parameters - canonical parameters. For any minimal surface of general type parameterized by canonical parameters we obtain Weierstrass representati…

We provide a parametric construction in terms of minimal surfaces of the Euclidean submanifolds of codimension two and arbitrary dimension that attain equality in an inequality due to De Smet, Dillen, Verstraelen and Vrancken. The latter involves the scalar curvature, the norm of the normal curvature tensor and the len…

Optimizes sharp curvature inequality on spheres, proving near-minimizers are close to standard metric.

We look at complete minimal surfaces of finite total curvature in $\mathbb{R}^4$. Similarly to the case of complex curves in $\mathbb{C}^2$ we introduce their {\it link at infinity}; we derive the {\it writhe number at infinity} which gives a formula for the total normal curvature of the surface. The knowledge of the l…

After appropriate normalizations an embedded disk whose second fundamental form has large norm contains a multi-valued graph, provided the L^P norm of the mean curvature is sufficiently small. This generalizes to non-minimal surfaces a well known result of Colding and Minicozzi.

Study on helicoidal singular minimal surfaces with specific properties.

Let $M$ be a space-like surface immersed in a 4-dimensional pseudo-Riemannian space form $R^4_2(c)$ with constant sectional curvature $c$ and index two. In the first part of this article, we prove that the Gauss curvature $K$, the normal curvature $K^D$, and mean curvature vector $H$ of $M$ satisfy the general inequali…

The study finds the minimum average area ratio on hyperbolic manifolds and its relation to scalar curvature.

The paper studies how submanifolds of a sphere evolve over time.

We prove that any strongly regular Weingarten surface in Euclidean space carries locally geometric principal parameters. The basic theorem states that any strongly regular Weingarten surface is determined up to a motion by its structural functions and the normal curvature function satisfying a geometric differential eq…

The study examines surfaces in isotropic space with specific Gauss map properties.

Yau's Affine Normal Descent optimizes smooth unconstrained problems with geometrically adapted directions.

In a previous work, we studied isoparametric functions on Riemannian manifolds, especially on exotic spheres. One result there says that, in the family of isoparametric hypersurfaces of a closed Riemannian manifold, there exist at least one minimal isoparametric hypersurface. In this note, we show such minimal isoparam…

Asymptotic net is an important concept in discrete differential geometry. In this paper, we show that we can associate affine discrete geometric concepts to an arbitrary non-degenerate asymptotic net. These concepts include discrete affine area, mean curvature, normal and co-normal vector fields and cubic form, and the…

We address the study of some curvature equations for distinguished submanifolds in para-Kähler geometry. We first observe that a para-complex submanifold of a para-Kähler manifold is minimal. Next we describe the extrinsic geometry of Lagrangian submanifolds in the para-complex Euclidean space D^n and discuss a number …

Minimal surfaces of general type in Euclidean 4-space are characterized with the conditions that the ellipse of curvature at any point is centered at this point and has two different principal axes. Any minimal surface of general type locally admits geometrically determined parameters - canonical parameters. In such pa…

We classify real hypersurfaces in complex space forms with constant principal curvatures and whose Hopf vector field has two nontrivial projections onto the principal curvature spaces. In complex projective spaces such real hypersurfaces do not exist. In complex hyperbolic spaces these are holomorphically congruent to …

We prove that the Gauss curvature and the curvature of the normal connection of any minimal surface in the four dimensional Euclidean space satisfy an inequality, which generates two classes of minimal surfaces: minimal surfaces of general type and minimal super-conformal surfaces. We prove a Bonnet-type theorem for st…

We study submanifolds whose principal curvatures, counted with multiplicities, do not depend on the normal direction. Such submanifolds, which we briefly call CPC submanifolds, are always austere, hence minimal, and have constant principal curvatures. Well-known classes of examples include totally geodesic submanifolds…

The study classifies timelike meridian surfaces in Minkowski 4-space.

In 3D space forms, a lens minimizes volume for a fixed surface area.

We consider the normalized Ricci flow evolving from an initial metric which is conformally compactifiable and asymptotically hyperbolic. We show that there is a unique evolving metric which remains in this class, and that the flow exists up to the time where the norm of the Riemann tensor diverges. Restricting to initi…

Constructs minimal surfaces over Pitot quadrilaterals using harmonic diffeomorphisms.

In this article a relation between curvature functionals for surfaces in the Euclidean space and area functionals in relative differential geometry will be given. Relative differential geometry can be described as the geometry of surfaces in the affine space, endowed with a distinguished "relative normal vector field" …