Submanifolds Of Euclidean Space Research Articles

Rotational Self-Shrinkers in Euclidean Spaces

The rotational embedded submanifold of $\mathbb{E}^{n+d}$ first studied by N. Kuiper. The special examples of this type are generalized Beltrami submanifolds and toroidals submanifold. The second named authour and at. all recently have considered $3-$dimensional rotational embedded submanifolds in $\mathbb{E}^{5}$. They gave some basic curvature properties of this type of submaifolds. Self-similar flows emerge as a special solution to the mean curvature flow that preserves the shape of the evolving submanifold. In this article we consider self-similar submanifolds in Euclidean spaces. We obtained some results related with self-shrinking rotational submanifolds in Euclidean $5-$space $\mathbb{E}^{5}$. Moreover, we give the necessary and sufficient conditions for these type of submanifolds to be homothetic solitons for their mean curvature flows.

Pythagorean Submanifolds in Euclidean Spaces

Pythagorean Submanifolds in Euclidean Spaces

Convergence of Martingales with Jumps on Submanifolds of Euclidean Spaces and its Applications to Harmonic Maps

Convergence of Martingales with Jumps on Submanifolds of Euclidean Spaces and its Applications to Harmonic Maps

Sobolev inequality and isoperimetric inequality for submanifolds in a smooth metric measure space

Brendle recently proved a sharp Sobolev inequality and logarithmic Sobolev inequality for submanifolds in Euclidean space. From the sharp Sobolev inequality, he achieved a breakthrough in the conjecture of isoperimetric inequality for minimal submanifolds. In this paper, we extend Brendle’s results to submanifolds in a smooth metric measure space. As an application, we prove some new isoperimetric-type inequalities in some smooth metric measure spaces. For example, we obtain a new isoperimetric-type inequality for self-expander.

On the focal locus of a submanifold in Euclidean space

On the focal locus of a submanifold in Euclidean space

A fractional Michael–Simon Sobolev inequality on convex hypersurfaces

The classical Michael-Simon and Allard inequality is a Sobolev inequality for functions defined on a submanifold of Euclidean space. It is governed by a universal constant independent of the manifold, but displays on the right-hand side an additional $L^p$ term weighted by the mean curvature of the underlying manifold. We prove here a fractional version of this inequality on hypersurfaces of Euclidean space that are boundaries of convex sets. It involves the Gagliardo semi-norm of the function, as well as its $L^p$ norm weighted by the fractional mean curvature of the hypersurface. As an application, we establish a new upper bound for the maximal time of existence in the smooth fractional mean curvature flow of a convex set. The bound depends on the perimeter of the initial set instead of on its diameter.

A manifold inexact augmented Lagrangian method for nonsmooth optimization on Riemannian submanifolds in Euclidean space

Abstract We develop a manifold inexact augmented Lagrangian framework to solve a family of nonsmooth optimization problem on Riemannian submanifold embedding in Euclidean space, whose objective function is the sum of a smooth function (but possibly nonconvex) and a nonsmooth convex function in Euclidean space. By utilizing the Moreau envelope, we get a smoothing Riemannian minimization subproblem at each iteration of the proposed method. Consequentially, each iteration subproblem is solved by a Riemannian Barzilai–Borwein gradient method. Theoretically, the convergence to critical point of the proposed method is established under some mild assumptions. Numerical experiments on compressed modes problems in physic and sparse principal component analysis demonstrate that the proposed method is a competitive method compared with some state-of-the-art methods.

Upper bounds for Steklov eigenvalues of submanifolds in Euclidean space via the intersection index

We obtain upper bounds for the Steklov eigenvalues σk(M) of a smooth, compact, n-dimensional submanifold M of Euclidean space with boundary Σ that involve the intersection indices of M and of Σ. One of our main results is an explicit upper bound in terms of the intersection index of Σ, the volume of Σ and the volume of M as well as dimensional constants. By also taking the injectivity radius of Σ into account, we obtain an upper bound that has the optimal exponent of k with respect to the asymptotics of the Steklov eigenvalues as k→∞.

The isoperimetric inequality for a minimal submanifold in Euclidean space

We prove a Sobolev inequality which holds on submanifolds in Euclidean space of arbitrary dimension and codimension. This inequality is sharp if the codimension is at most 2 2 . As a special case, we obtain a sharp isoperimetric inequality for minimal submanifolds in Euclidean space of codimension at most 2 2 .

On the first eigenvalue of the Laplace operator for compact spacelike submanifolds in Lorentz–Minkowski spacetime 𝕃m

By means of a counter-example, we show that the Reilly theorem for the upper bound of the first non-trivial eigenvalue of the Laplace operator of a compact submanifold of Euclidean space (Reilly, 1977, Comment. Mat. Helvetici, 52, 525–533) does not work for a (codimension ⩾2) compact spacelike submanifold of Lorentz–Minkowski spacetime. In the search of an alternative result, it should be noted that the original technique in (Reilly, 1977, Comment. Mat. Helvetici, 52, 525–533) is not applicable for a compact spacelike submanifold of Lorentz–Minkowski spacetime. In this paper, a new technique, based on an integral formula on a compact spacelike section of the light cone in Lorentz–Minkowski spacetime is developed. The technique is genuine in our setting, that is, it cannot be extended to another semi-Euclidean spaces of higher index. As a consequence, a family of upper bounds for the first eigenvalue of the Laplace operator of a compact spacelike submanifold of Lorentz–Minkowski spacetime is obtained. The equality for one of these inequalities is geometrically characterized. Indeed, the eigenvalue achieves one of these upper bounds if and only if the compact spacelike submanifold lies minimally in a hypersphere of certain spacelike hyperplane. On the way, the Reilly original result is reproved if a compact submanifold of a Euclidean space is naturally seen as a compact spacelike submanifold of Lorentz–Minkowski spacetime through a spacelike hyperplane.

Positive solutions to Schrödinger equations and geometric applications

Abstract A variant of Li–Tam theory, which associates to each end of a complete Riemannian manifold a positive solution of a given Schrödinger equation on the manifold, is developed. It is demonstrated that such positive solutions must be of polynomial growth of fixed order under a suitable scaling invariant Sobolev inequality. Consequently, a finiteness result for the number of ends follows. In the case when the Sobolev inequality is of particular type, the finiteness result is proven directly. As an application, an estimate on the number of ends for shrinking gradient Ricci solitons and submanifolds of Euclidean space is obtained.

The Logarithmic Sobolev Inequality for a Submanifold in Euclidean Space

AbstractWe prove a sharp logarithmic Sobolev inequality that holds for submanifolds in Euclidean space of arbitrary dimension and codimension. Like the Michael‐Simon Sobolev inequality, this inequality includes a term involving the mean curvature. © 2020 Wiley Periodicals, Inc.

Warped product submanifolds in metallic Riemannian manifolds

In this paper, we study the existence of proper warped product submanifolds in metallic (or Golden) Riemannian manifolds and we discuss about semi-invariant, semi-slant and, respectively, hemi-slant warped product submanifolds in metallic and Golden Riemannian manifolds. Also, we provide some examples of warped product submanifolds in Euclidean spaces.

On the Extrinsic Principal Directions and Curvatures of Lagrangian Submanifolds

From the basic geometry of submanifolds will be recalled what are the extrinsic principal tangential directions, (first studied by Camille Jordan in the 18seventies), and what are the principal first normal directions, (first studied by Kostadin Trenčevski in the 19nineties), and what are their corresponding Casorati curvatures. For reasons of simplicity of exposition only, hereafter this will merely be done explicitly in the case of arbitrary submanifolds in Euclidean spaces. Then, for the special case of Lagrangian submanifolds in complex Euclidean spaces, the natural relationships between these distinguished tangential and normal directions and their corresponding curvatures will be established.

Ancient solutions to mean curvature flow for isoparametric submanifolds

Mean curvature flow for isoparametric submanifolds in Euclidean spaces and spheres was studied in Liu and Terng (Duke Math J 147(1):157–179, 2009). In this paper, we will show that all these solutions are ancient solutions and study their limits as time goes to negative infinity. We also discuss rigidity of ancient mean curvature flows for hypersurfaces in spheres and its relation to the Chern’s conjecture on the norm of the second fundamental forms of minimal hypersurfaces in spheres.

Reilly-Type Inequalities for Paneitz and Steklov Eigenvalues

We prove Reilly-type upper bounds for different types of eigenvalue problems on submanifolds of Euclidean spaces with density. This includes the eigenvalues of Paneitz-like operators as well as three types of generalized Steklov problems. In the case without density, the equality cases are discussed and we prove some stability results for hypersurfaces which derive from a general pinching result about the moment of inertia.

Chord shortening flow and a theorem of Lusternik and Schnirelmann

We introduce a new geometric flow called the chord shortening flow which is the negative gradient flow for the length functional on the space of chords with end points lying on a fixed submanifold in Euclidean space. As an application, we give a simplified proof of a classical theorem of Lusternik and Schnirelmann (and a generalization by Riede and Hayashi) on the existence of multiple orthogonal geodesic chords. For a compact convex planar domain, we show that any convex chord which is not orthogonal to the boundary would shrink to a point in finite time under the flow.

Rotational submanifolds in Euclidean spaces

The rotational embedded submanifold was first studied by Kuiper as a submanifold in [Formula: see text]. The generalized Beltrami submanifolds and toroidal submanifold are the special examples of these kind of submanifolds. In this paper, we consider [Formula: see text]-dimensional rotational embedded submanifolds in Euclidean [Formula: see text]-space [Formula: see text]. We give some basic curvature properties of this type of submanifolds. Further, we obtain some results related with the scalar curvature and mean curvature of these submanifolds. As an application, we give an example of rotational submanifold in [Formula: see text].

Hypersurfaces with Generalized 1-Type Gauss Maps

In this paper, we study submanifolds in a Euclidean space with a generalized 1-type Gauss map. The Gauss map, G, of a submanifold in the n-dimensional Euclidean space, En, is said to be of generalized 1-type if, for the Laplace operator, Δ, on the submanifold, it satisfies ΔG=fG+gC, where C is a constant vector and f and g are some functions. The notion of a generalized 1-type Gauss map is a generalization of both a 1-type Gauss map and a pointwise 1-type Gauss map. With the new definition, first of all, we classify conical surfaces with a generalized 1-type Gauss map in E3. Second, we show that the Gauss map of any cylindrical surface in E3 is of the generalized 1-type. Third, we prove that there are no tangent developable surfaces with generalized 1-type Gauss maps in E3, except planes. Finally, we show that cylindrical hypersurfaces in En+2 always have generalized 1-type Gauss maps.

The value distribution under the Gauss map for a minimal submanifold

In this paper, some new aspects for the value distributionof the image under the Gauss map for a minimal submanifold in Euclidean space are reviewed, includingYaus problem for minimal hypersurfaces and Lawson-Ossermans problem for higher codimension. Among them theauthors joint work with his collaborators in recent years is described.