Related papers: Wintgen inequality for statistical surfaces
We present a complete system of inequalities for the inradius, circumradius, and diameter in the $3$-dimensional Euclidean space. To do so, we prove quasiconcavity of the inradius evaluated over $n$-simplices with a common facet…
Thin films or sheets subjected to external forces often undergo mechanical instability, leading to regular patterns of wrinkles, folds, and creases. As can be anticipated from the difficulty of flattening a curved globe, any natural…
In this paper, we prove gap results for constant mean curvature (CMC) surfaces. Firstly, we find a natural inequality for CMC surfaces which imply convexity for distance function. We then show that if $\Sigma$ is a complete, properly…
In this paper, we establish two families of sharp geometric inequalities for closed hypersurfaces in space forms or other warped product manifolds. Both families of inequalities compare three distinct geometric quantities. The first family…
The paper investigates higher dimensional analogues of Burago's inequality bounding the area of a closed surface by its total curvature. We obtain sufficient conditions for hypersurfaces in 4-space that involve the Ricci curvature. We get…
In this paper, we investigate the mean curvature flow of compact surfaces in $4$-dimensional space forms. We prove the convergence theorems for the mean curvature flow under certain pinching conditions involving the normal curvature, which…
Wintgen ideal surfaces in E^4 form an important family of surfaces, namely surfaces with circular ellipse of curvature. Obviously, Wintgen ideal surfaces satisfy the pointwise equality K+K_N=H^2. In the present study we consider the Wintgen…
In this work, complete constant mean curvature 1 (CMC-1) surfaces in hyperbolic 3-space with total absolute curvature at most 4 pi are classified. This classification suggests that the Cohn-Vossen inequality can be sharpened for surfaces…
A study of the generalized Weierstrass system which can be used to induce mean curvature surfaces in three-dimensional Euclidean space is presented.
In this paper we study curvature types of immersed surfaces in three-dimensional (normed or) Minkowski spaces. By endowing the surface with a normal vector field, which is a transversal vector field given by the ambient Birkhoff…
Considering the tangent plane at a point to a surface in the four-dimensional Euclidean space, we find an invariant of a pair of two tangents in this plane. If this invariant is zero, the two tangents are said to be conjugate. When the two…
We investigate some characteristic properties of specific Weingarten surfaces in the three-dimensional Euclidean space using the nets of the lines of curvature resp. the asymptotic lines on both central surfaces of them.
The gauge invariant elastic metric on the shape space of surfaces involves the mean curvature and the normal deformation, i.e. the sum and the difference of the principal curvatures $\kappa_1,\kappa_2$. The proposed gauge invariant elastic…
It was proved by Graham and Witten in 1999 that conformal invariants of submanifolds can be obtained via volume renormalization of minimal surfaces in conformally compact Einstein manifolds. The conformal invariant of a submanifold $\Sigma$…
This essay, an excerpt of the author's Ph.D. in Philosophy of mathematics (2012) thought of as being a companion to recent discoveries of new explicit Cartan geometry curvatures, analyzes how Gauss, after having devised the isometrically…
In the present paper, we propose a new discrete surface theory on 3-valent embedded graphs in the 3-dimensional Euclidean space which are not necessarily discretization or approximation of smooth surfaces. The Gauss curvature and the mean…
The total mean curvature functional for submanifolds into the Riemannian product space $\mathbb{S}^n\times\mathbb{R}$ is considered and its first variational formula is presented. Later on, two second order differential operators are…
We study the invariants of surfaces in 4-manifolds extracted from the Seiberg-Witten and the Ozsvath-Szabo invariants of their fiber sums with auxiliary Lefschetz fibrations. Such invariants involve relative Spin_c structures and can be…
It is shown that the nature of quantum statistics can study in assumption of existence of a background of random gravitational fields and waves, distributed isotropically in the space. This background is capable of correlating phases of…
This paper is devoted to the Moser-Trudinger inequality on smooth riemanniansurfaces. We establish that the constants involved can be chosen to depend on only 3parameters, which are the systole, isoperimetric constant and curvature of the…