MONOTONICITY FORMULAE, VANISHING THEOREMS AND SOME GEOMETRIC APPLICATIONS

Using Hilbert’s criterion, we consider the stress-energy tensor associated to the bienergy functional. We show that it derives from a variational problem on metrics and exhibit the peculiarity of dimension four. First, we use this tensor to construct new examples of biharmonic maps, then classify maps with vanishing or parallel stress-energy tensor and Riemannian immersions whose stress-energy tensor is proportional to the metric, thus obtaining a weaker but high-dimensional version of the Hopf Theorem on compact constant mean curvature immersions. We also relate the stress-energy tensor of the inclusion of a submanifold in Euclidean space with the harmonic stress-energy tensor of its Gauss map.

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.

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.

Let M be a properly immersed n-dimensional complete minimal submanifold in Euclidean space Rn+p of dimension n+p. Let A be the second fundamental form of the immersion, and r the extrinsic distance from the origin. Suppose M has one end and inft supr(x)>t r2(x) |A|2(x) 1.

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 .

In this paper, we study n-dimensional complete immersed submanifolds in a Euclidean space En+p. We prove that if Mn is an n-dimensional compact connected immersed submanifold with nonzero mean curvature H in En+p and satisfies either: (1)S≤n2H2n-1, or (2)n2H2≤(n-1)Rn-2, then Mn is diffeomorphic to a standard n-sphere, where S and R denote the squared norm of the second fundamental form of Mn and the scalar curvature of Mn, respectively. On the other hand, in the case of constant mean curvature, we generalized results of Klotz and Osserman [11] to arbitrary dimensions and codimensions; that is, we proved that the totally umbilical sphere Sn(c), the totally geodesic Euclidean space En, and the generalized cylinder Sn-1(c)×E1 are only n-dimensional (n>2) complete connected submanifolds Mn with constant mean curvature H in En+p if S≤n2H2/(n-1) holds.

This paper studies minimal surfaces in Kähler manifolds of constant holomorphic sectional curvature using the technique of the moving frame. In particular, we provide a classification of the minimal two-spheres in C P n {\mathbf {C}}{P^n} , complex projective n n -space, equipped with the Fubini-Study metric. This classification can be described as follows: To each holomorphic curve in C P n {\mathbf {C}}{P^n} classically there is associated a particular framing of C n + 1 {{\mathbf {C}}^{n + 1}} called the Frenet frame. Each element of the Frenet frame induces a minimal surface in C P n {\mathbf {C}}{P^n} . The classification theorem states that all minimal surfaces of topological type of the two-sphere occur in this manner. The theorem is proved using holomorphic differentials that occur naturally on minimal surfaces in Kähler manifolds of constant holomorphic sectional curvature together with the Riemann-Roch Theorem.

In this paper, we investigate minimal submanifolds in Euclidean space with positive index of relative nullity. Let \(M^m\) be a complete Riemannian manifold and let \(f:M^m\rightarrow \mathbb {R}^n\) be a minimal isometric immersion with index of relative nullity at least \(m-2\) at any point. We show that if the Omori–Yau maximum principle for the Laplacian holds on \(M^m\), for instance, if the scalar curvature of \(M^m\) does not decrease to \(-\infty \) too fast or if the immersion f is proper, then the submanifold must be a cylinder over a minimal surface.

We study a complete noncompact minimal submanifold Mn in a sphere Sn+p. We prove there is no nontrivial L2 harmonic 1-form and at most one nonparabolic end on M if the total curvature is bounded from above by a constant depending only on n. The rigidity theorem is a generalized version of Ni’s, Yun’s and the second author’s results on submanifolds in Euclidean spaces and Seo’s result on minimal submanifolds in hyperbolic spaces.

The object of the paper is to study some compact submanifolds in the Euclidean space Rn whose mean curvature vector is parallel in the normal bundle. First we prove that there does not exist an n‐dimensional compact simply connected totally real submanifold in R2n whose mean curvature vector is parallel. Then we show that the n‐dimensional compact totally real submanifolds of constant curvature and parallel mean curvature in R2n are flat. Finally we show that compact Positively curved submanifolds in Rn with parallel mean curvature vector are homology spheres. The last result in particular for even dimensional submanifolds implies that their Euler poincaré characteristic class is positive, which for the class of compact positively curved submanifolds admiting isometric immersion with parallel mean curvature vector in Rn, answers the problem of Chern and Hopf

Let ]]> ]]> ]]> ]]> ]]> ]]> ]]> ]]> ]]> ]]> ]]> ]]> ]]> ]]> ]]> ]]> ]]> M^n$ be a Riemannian $n$-manifold with $n\ge 4$. Consider the Riemannian invariant $\sigma(2)$ defined by ]]> ]]> \sigma(2)=\tau-\frac{(n-1)\min \Ric}{n^2-3n+4}, $$ where $\tau$ is the scalar curvature of $M^n$ and $(\min \Ric)(p)$ is the minimum of the Ricci curvature of $M^n$ at $p$. In an earlier article, B. Y. Chen established the following sharp general inequality: $$ \sigma(2)\le \frac{n^2{(n-2)}^2}{2(n^2-3n+4)}H^2 $$ for arbitrary $n$-dimensional conformally flat submanifolds in a Euclidean space, where $H^2$ denotes the squared mean curvature. The main purpose of this paper is to completely classify the extremal class of conformally flat submanifolds which satisfy the equality case of the above inequality. Our main result states that except open portions of totally geodesic $n$-planes, open portions of spherical hypercylinders and open portion of round hypercones, conformally flat submanifolds satifying the equality case of the inequality are obtained from some loci of $(n-2)$-spheres around some special coordinate-minimal surfaces.

The topology of the set of singular support hyperplanes and hyperspheres to a smooth submanifold in Euclidean space is studied. As a corollary, some relations between differential-geometric characteristics of a manifold are obtained. In particular, if a simple closed embedded generic curve in a plane has C global vertices (where the curvature circles are support circles to the curve) and T support circles touching the curve at three points, then C − T = 4. Similar invariants are also obtained for submanifolds in higher-dimensional spaces.

We study properties of principal curvature vectors of normally flat Ric-semisymmetric submanifolds in Euclidean spaces and give a geometric description of two particular classes of such submanifolds.

In this paper we study the structure of an immersed submanifold Mn in a Riemannian manifold with flat normal bundle in two ways. Firstly, we prove that if Mn is compact and satisfies some pointwise pinching condition, and assume further that the ambient space has pure curvature tensor and non-negative isotropic curvature, then the Betti numbers βp(M) = 0 for 2 ≤ p ≤ n−2. Secondly, suppose that Mn is a complete non-compact submanifold in the Euclidean space with finite total curvature in the sense that its traceless second fundament form has finite Ln-norm, then we show that the spaces of L2 harmonic p-forms on Mn have finite dimensions for all 2 ≤ p ≤ n−2.

The study of biharmonic submanifolds in Euclidean spaces was introduced in the middle of the 1980s by the author in his program studying finite-type submanifolds. He defined biharmonic submanifolds in Euclidean spaces as submanifolds whose position vector field (x) satisfies the biharmonic equation, i.e., Δ2x=0. A well-known conjecture proposed by the author in 1991 on biharmonic submanifolds states that every biharmonic submanifold of a Euclidean space is minimal, well known today as Chen’s biharmonic conjecture. On the other hand, independently, G.-Y. Jiang investigated biharmonic maps between Riemannian manifolds as the critical points of the bi-energy functional. In 2002, R. Caddeo, S. Montaldo, and C. Oniciuc pointed out that both definitions of biharmonicity of the author and G.-Y. Jiang coincide for the class of Euclidean submanifolds. Since then, the study of biharmonic submanifolds and biharmonic maps has attracted many researchers, and many interesting results have been achieved. A comprehensive survey of important results on this conjecture and on many related topics was presented by Y.-L. Ou and B.-Y. Chen in their 2020 book. The main purpose of this paper is to provide a detailed survey of recent developments in those subjects after the publication of Ou and Chen’s book.