Some Results on Calibrated Submanifolds in Euclidean Space of Cohomogeneity One and Two

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.

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 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.

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 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.

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.

Using the stress energy tensor, we establish some monotonicity formulae for vector bundle-valued p-forms satisfying the conservation law, provided that the base Riemannian (resp. K\ahler) manifolds poss some real (resp. complex) p-exhaustion functions. Vanishing theorems follow immediately from the monotonicity formulae under suitable growth conditions on the energy of the p-forms. As an application, we establish a monotonicity formula for the Ricci form of a K\ahler manifold of constant scalar curvature and then get a growth condition to derive the Ricci flatness of the K\ahler manifold. In particular, when the curvature does not change sign, the K\ahler manifold is isometrically biholomorphic to C^m. Another application is to deduce the monotonicity formulae for volumes of minimal submanifolds in some outer spaces with suitable exhaustion functions. In this way, we recapture the classical volume monotonicity formulae of minimal submanifolds in Euclidean spaces. We also apply the vanishing theorems to Bernstein type problem of submanifolds in Euclidean spaces with parallel mean curvature. In particular, we may obtain Bernstein type results for minimal submanifolds, especially for minimal real K\ahler submanifolds under weaker conditions.

Ruled austere submanifolds of dimension four

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.

We 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.

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.

A martingale characterization of Brownian motion on a submanifold of Euclidean space is proved and the implications of the consequent martingale representation are discussed.

In this paper the issue of filtering and smoothing in continuous discrete time is studied when the state variable evolves in some submanifold of Euclidean space, which may not have the usual Lebesgue measure. Formal expressions for prediction and smoothing problems are reviewed, which agree with the classical results except that the formal adjoint of the generator is different in general. These results are used to generalise the projection approach to filtering and smoothing to the case when the state variable evolves in some submanifold that lacks a Lebesgue measure. The approach is used to develop projection filters and smoothers based on the von Mises–Fisher distribution, which are shown to be outperform Gaussian estimators both in terms of estimation accuracy and computational speed in simulation experiments involving the tracking of a gravity vector.