Thomas Hasanis

Bio: Thomas Hasanis is an academic researcher from University of Ioannina. The author has contributed to research in topics: Mean curvature & Curvature. The author has an hindex of 9, co-authored 31 publications receiving 202 citations.

A pinching theorem for minimal hypersurfaces in a sphere

Abstract: We study compact minimal hypersurfaces M n in \(S^{n+1}\) with two distinct principal curvatures and prove that if the squared norm S of the second fundamental form of M n satisfies \(S \geqq n\), then \(S \equiv n\) and M n is a minimal Clifford torus.

Hypersurfaces with constant scalar curvature and constant mean curvature

Abstract: Non-spherical hypersurfaces in E 4 with non-zero constant mean curvature and constant scalar curvature m'e the only hypersurfaces possessing the following property: Its position vector can be written as a sum of two non-constant maps, which are eigenmaps of the Laplacian operator with corresponding eigenvalues the zero and a non-zero constant.

Classification and Construction of Minimal Translation Surfaces in Euclidean Space

Abstract: A translation surface of Euclidean space $${\mathbb {R}}^3$$ is the sum of two regular curves $$\alpha $$ and $$\beta $$, called the generating curves. In this paper we classify the minimal translation surfaces of $${\mathbb {R}}^3$$ and we give a method of construction of explicit examples. Besides the plane and the minimal surfaces of Scherk type, it is proved that up to reparameterizations of the generating curves, any minimal translation surface is described as $$\Psi (s,t)=\alpha (s)+\alpha (t)$$, where $$\alpha $$ is a curve parameterized by arc length s, its curvature $$\kappa $$ is a positive solution of the autonomous ODE $$(y')^2+y^4+c_3y^2+c_1^2y^{-2}+c_1c_2=0$$ and its torsion is $$\tau (s)=c_1/\kappa (s)^2$$. Here $$c_1 ot =0$$, $$c_2$$ and $$c_3$$ are constants such that the cubic equation $$-\lambda ^3+c_2\lambda ^2-c_3\lambda +c_1=0$$ has three real roots $$\lambda _1$$, $$\lambda _2$$ and $$\lambda _3$$.

Ricci curvature and minimal submanifolds

Abstract: The general question that served as the starting point for this paper was to find necessary conditions on those Riemannian metrics that arise as the induced metrics on minimal hypersurfaces or submanifolds of hyperspheres of a Euclidean space. There is an abundance of complete minimal hypersurfaces in the unit hypersphere Sn+1. We recall some well known examples. Let Sm(r) = {x ∈ Rn+1, |x| = r}, Sn−m(s) = {y ∈ Rn−m+1, |y| = s}, where r and s are positive numbers with r2+ s2 = 1; then Sm(r)×Sn−m(s) = {(x, y) ∈ Rn+2, x ∈ Sm(r), y ∈ Sn−m(s)} is a hypersurface of the unit hypersphere in Rn+2. As is well known, this hypersurface has two distinct constant principal curvatures: One is s/r of multiplicity m, the other is −r/s of multiplicity n − m. This hypersurface is called a Clifford hypersurface. Moreover, it is minimal only in the case r = √ m/n, s = √ (n − m)/n and is called a Clifford minimal hypersurface. Otsuki [11] proved that if Mn is a compact minimal hypersurface in Sn+1 with two distinct principal curvatures of multiplicity greater than 1, then Mn is a Clifford minimal hypersurface Sm( √ m/n) × Sn−m(√(n − m)/n), 1 < m < n − 1. Furthermore, Otsuki constructed infinitely many compact minimal hypersurfaces, other than S1( √ 1/n)× Sn−1(√(n − 1)/n), with two distinct principal curvatures and one of them be simple, which are not congruent to each other in Sn+1. Using the method of equivariant Differential Geometry, families of infinitely many nonequitorial minimal embeddings of the n-sphere were constructed in Sn+1 for the dimensions n = 3, 4, 5, 6, 7, 9, 11 and 13 by Hsiang and Sterling in [7]. Moreover, the same method will also produce at least one nonequitorial in all even dimensional spheres ([16]). These results settled the so called spherical Bernstein problem posed by S.S. Chern. Let Mn be an n-dimensional smooth and oriented Riemannian manifold, and f : Mn → Sn+1 an isometric immersion of Mn into the unit hypersphere Sn+1 of Rn+2. The unit normal vectorfield of f in Sn+1 induces a map

Minimal hypersurfaces with zero Gauss-Kronecker curvature

Abstract: We investigate complete minimal hypersurfaces in the Euclidean space ${R}^{4}$, with Gauss-Kronecker curvature identically zero. We prove that, if $f:M^{3}\rightarrow {R}^{4}$ is a complete minimal hypersurface with Gauss-Kronecker curvature identically zero, nowhere vanishing second fundamental form and scalar curvature bounded from below, then $f(M^{3})$ splits as a Euclidean product $L^{2}\times {R}$, where $L^{2}$ is a complete minimal surface in $ {R}^{3}$ with Gaussian curvature bounded from below.

An estimate for the curvature of bounded submanifolds

Abstract: 1. Statement of the result. All manifolds considered in this paper shall be connected, of class C"' (smooth), and dimension at least 2. Im- mersions will also be smooth, and of codimension at least 1. If M and M are Riemannian manifolds and (p: M - M is an isometric immersion with the property that sp(M) lies in a ball, we intend to estimate the supremum of the sectional curvature K of M in terms of the sectional curvature K of M and the radius of the ball. This is our result: THEOREM A. Let Ml' be a complete Riemannian manifold whose scalar curvature is bounded below; let M" +q be a Riemannian manifold with q c n -1, and Bx a closed normal ball in Mt4+q, of radius X. Sup- pose Sp:M" - j M"+q is an isometric immersion with the property that

Some open problems and conjectures on submanifolds of finite type: recent development

Abstract: Submanifolds of finite type were introduced by the author during the late 1970s. The first results on this subject were collected in author's books [26,29]. In 1991, a list of twelve open problems and three conjectures on finite type submanifolds was published in [40]. A detailed survey of the results, up to 1996, on this subject was given by the author in [48]. Recently, the study of finite type submanifolds, in particular, of biharmonic submanifolds, have received a growing attention with many progresses since the beginning of this century. In this article, we provide a detailed account of recent development on the problems and conjectures listed in [40].

Mathematical Aspects of Design of Beam Shaping Surfaces in Geometrical Optics

Abstract: Numerous optical and electromagnetic applications require synthesis of reflecting and refracting surfaces capable of reshaping the energy radiation intensity of a given source into a prescribed output irradiance distribution. Determination of such surfaces requires investigation of nonlinear, second order partial differential equations of Monge-Ampere type and development of computational algorithms for constructing their numerical solutions. These equations are very far from being standard and it is quite remarkable that geometric ideas not only provide natural means for their analysis but also means for computing solutions numerically. In this paper we survey some of these problems and describe the current progress in their study.

Some open problems and conjectures on submanifolds of finite type: recent development

Abstract: Submanifolds of finite type were introduced by the author during the late 1970s. The first results on this subject were collected in author's books [26,29]. In 1991, a list of twelve open problems and three conjectures on finite type submanifolds was published in [40]. A detailed survey of the results, up to 1996, on this subject was given by the author in [48]. Recently, the study of finite type submanifolds, in particular, of biharmonic submanifolds, have received a growing attention with many progresses since the beginning of this century. In this article, we provide a detailed account of recent development on the problems and conjectures listed in [40].

Constant mean curvature hypersurfaces in warped product spaces

Abstract: We study hypersurfaces of constant mean curvature immersed into warped product spaces of the form , generalizing previous results by Montiel. We also extend a result of Guan and Spruck from hyperbolic ambient space to the general situation of warped products. This extension allows us to give a slightly more general version of a result by Montiel and to derive height estimates for compact constant mean curvature hypersurfaces with boundary in a leaf.