Submanifolds In Riemannian Manifolds Research Articles

Optimal inequalities for submanifolds of Riemann manifolds of nearly quasi-constant curvature with quarter-symmetric connection

In this paper, we obtain Chen inequalities for submanifolds in Riemannian manifolds of nearly quasi-constant curvature with a special kind of quartersymmetric connection and discuss the equality case of the inequalities. We also obtain some Casorati inequalities for submanifolds in Riemannian manifolds of nearly quasi-constant curvature with the quarter-symmetric connection.

Geometry of k-Yamabe Solitons on Euclidean Spaces and Its Applications to Concurrent Vector Fields

In this paper, we give some classifications of the k-Yamabe solitons on the hypersurfaces of the Euclidean spaces from the vector field point of view. In several results on k-Yamabe solitons with a concurrent vector field on submanifolds in Riemannian manifolds, is proved that a k-Yamabe soliton (Mn,g,vT,λ) on a hypersurface in the Euclidean space Rn+1 is contained either in a hypersphere or a hyperplane. We provide an example to support this study and all of the results in this paper can be implemented to Yamabe solitons for k-curvature with k=1.

Classification of almost Yamabe solitons in Euclidean spaces

In this paper, we completely classify almost Yamabe solitons on hypersurfaces in Euclidean spaces arisen from the position vector field. Some results of almost Yamabe solitons with a concurrent vector field and almost Yamabe solitons on submanifolds in Riemannian manifolds equipped with a concurrent vector field are also presented. Moreover, we classify complete Ricci solitons on minimal submanifolds in non-positively curved space forms. For almost Yamabe solitons, all of the results in this paper can be applied to Yamabe solitons.

Curvature estimates for stable free boundary minimal hypersurfaces

Abstract In this paper, we prove uniform curvature estimates for immersed stable free boundary minimal hypersurfaces satisfying a uniform area bound, which generalize the celebrated Schoen–Simon–Yau interior curvature estimates up to the free boundary. Our curvature estimates imply a smooth compactness theorem which is an essential ingredient in the min-max theory of free boundary minimal hypersurfaces developed by the last two authors. We also prove a monotonicity formula for free boundary minimal submanifolds in Riemannian manifolds for any dimension and codimension. For 3-manifolds with boundary, we prove a stronger curvature estimate for properly embedded stable free boundary minimal surfaces without a-priori area bound. This generalizes Schoen’s interior curvature estimates to the free boundary setting. Our proof uses the theory of minimal laminations developed by Colding and Minicozzi.

On the structure and volume growth of submanifolds in Riemannian manifolds

The aim of this note is two-fold. First, we investigate the relations between the volume growth of a submanifold and its second fundamental form. In the second part, we discuss the relations between the index of some Schrödinger operators and the structure of a submanifold, and prove some one-end theorems.

Lqbounds on restrictions of spectral clusters to submanifolds for low regularity metrics

We prove $L^q$ bounds on the restriction of spectral clusters to submanifolds in Riemannian manifolds equipped with metrics of $C^{1,\alpha}$ regularity for $0 \leq \alpha \leq 1$. Our results allow for Lipschitz regularity when $\alpha =0$, meaning they give estimates on manifolds with boundary. When $0< \alpha \leq 1$, the scalar second fundamental form for a codimension 1 submanifold can be defined, and we show improved estimates when this form is negative definite. This extends results of Burq-G\'erard-Tzvetkov and Hu to manifolds with low regularity metrics.

Isoperimetric inequalities for minimal submanifolds in Riemannian manifolds: a counterexample in higher codimension

For compact Riemannian manifolds with convex boundary, B.White proved the following alternative: Either there is an isoperimetric inequality for minimal hypersurfaces or there exists a closed minimal hypersurface, possibly with a small singular set. There is the natural question if a similar result is true for submanifolds of higher codimension. Specifically, B.White asked if the non-existence of an isoperimetric inequality for k-varifolds implies the existence of a nonzero, stationary, integral k-varifold. We present examples showing that this is not true in codimension greater than two. The key step is the construction of a Riemannian metric on the closed four-dimensional ball B with the following properties: (1) B has strictly convex boundary. (2) There exists a complete nonconstant geodesic. (3) There does not exist a closed geodesic in B.

Pullback of measures on Riemannian manifolds

A general formula for the pullback of measures on submanifolds of Riemannian manifolds is derived. We prove the equivalence of three different approaches to distributions supported on submanifolds, originating from Gelfand and Shilov, Friedlander, and a characterization using single-layer distributions, respectively.

Interface Foliation near Minimal Submanifolds in Riemannian Manifolds with Positive Ricci Curvature

Let $${(\mathcal {M},\tilde{g})}$$ be an N-dimensional smooth compact Riemannian manifold. We consider the singularly perturbed Allen–Cahn equation $$\varepsilon ^2 \Delta _{\tilde g} u \, + \, (1 - u^2 )u\, =\, 0 \quad {\rm{in}} \, \mathcal {M},$$ where $${\varepsilon}$$ is a small parameter. Let $${{\mathcal {K} \subset \mathcal {M}}}$$ be an (N - 1)-dimensional smooth minimal submanifold that separates $${\mathcal{M}}$$ into two disjoint components. Assume that $${\mathcal{K}}$$ is nondegenerate in the sense that it does not support non-trivial Jacobi fields, and that $${{|A_\mathcal {K} |^2 + {\rm {Ric}}_{\tilde g} (v _{\mathcal K}, v_{\mathcal K})}}$$ is positive along $${\mathcal{K}}$$ . Then for each integer m ≥ 2, we establish the existence of a sequence $${\varepsilon = \varepsilon{_j} \rightarrow 0}$$ , and solutions $${u_\varepsilon}$$ with m-transition layers near $${\mathcal{K}}$$ , with mutual distance $${O(\varepsilon |\, {\rm {ln}}\, \varepsilon|)}$$ .

Minimality in CR geometry and the CR Yamabe problem on CR manifolds with boundary

We study the minimality of an isometric immersion of a Riemannian manifold into a strictly pseudoconvex CR manifold M endowed with the Webster metric hence consider a version of the CR Yamabe problem for CR manifolds with boundary. This occurs as the Yamabe problem for the Fefferman metric (a Lorentzian metric associated to a choice of contact structure θ on M , [20]) on the total space of the canonical circle bundle S 1 →C(M) → π M (a manifold with boundary ∂C(M)= π −1 (∂M) and is shown to be a nonlinear subelliptic problem of variational origin. For any real surface N={φ=0}⊂ H 1 we show that the mean curvature vector of N⇀ H 1 is expressed by H=− 1 2 ∑ j=1 2 X j (|Xφ | −1 X j φ)ξ provided that N is tangent to the characteristic direction T of ( H 1 , θ 0 ) , thus demonstrating the relationship between the classical theory of submanifolds in Riemannian manifolds (cf. e.g.[7]) and the newer investigations in [1], [6], [8] and [16]. Given an isometric immersion Ψ:N→ H n of a Riemannian manifold into the Heisenberg group we show that ΔΨ=2J T ⊥ hence start a Weierstrass representation theory for minimal surfaces in H n .

On $$(k,\varepsilon)$$ -saddle submanifolds of Riemannian manifolds

The paper continues our (in collaboration with A. Borisenko [J. Differential Geom. Appl. 20 p., to appear]) discovery of the new classes of \((k,\varepsilon)\)-saddle, \((k,\varepsilon)\)-parabolic, and \((k,\varepsilon)\)-convex submanifolds ( \(\,\varepsilon\ge0\)). These are defined in terms of the eigenvalues of the 2nd fundamental forms of each unit normal of the submanifold, extending the notion of k-saddle, k-parabolic, k-convex submanifolds ( \(\,\varepsilon=0\)). It follows that the definition of \((k,\varepsilon)\)-saddle submanifolds is equivalent to the existence of \(\varepsilon\)-asymptotic subspaces in the tangent space. We prove non-embedding theorems of \((k,\varepsilon)\)-saddle submanifolds, theorems about 1-connectedness and homology groups of these submanifolds in Riemannian spaces of positive (sectional or qth Ricci) curvature, in particular, spherical and projective spaces. We apply these results to submanifolds with ‘small’ normal curvature, \(k_n\le\varepsilon\), and for submanifolds with extrinsic curvature \(\le\varepsilon^2\) (resp., non-positive) and small codimension, and include some illustrative examples. The results of the paper generalize theorems about totally geodesic, minimal and k-saddle submanifolds by Frankel; Borisenko, Rabelo and Tenenblat; Kenmotsu and Xia; Mendonca and Zhou.

On the Mean Exit Time for Compact Symmetric Spaces

Given a compact symmetric space, M, we obtain the mean exit time function from a principal orbit, for a Brownian particle starting and moving in a generalized ball whose boundary is the principal orbit. We also obtain the mean exit time function of a tube of radius r around special totally geodesic submanifolds P of M. Finally we give a comparison result for the mean exit time function of tubes around submanifolds in Riemannian manifolds, using these totally geodesic submanifolds in compact symmetric spaces as a model.

On Saddle Submanifolds of Riemannian Manifolds

Saddle submanifolds are considered. A characterization of such submanifolds of Euclidean space is given in terms of sectional curvature. Extending results of T. Frankel, K. Kenmotsu and C. Xia, we determine under what conditions two complete saddle submanifolds of a complete connected Riemannian manifold M, with nonnegative k-Ricci curvature, must intersect. Moreover, if M has positive k -Ricci curvature and the dimension of a compact saddle submanifold satisfies a certain inequality then we show that the homomorphism of the fundamental groups π1(M) and π1(M) is surjective.

On closed minimal submanifolds in pinched Riemannian manifolds

In this paper, we first prove a generalized Simons integral inequality for closed minimal submanifolds in a Riemannian manifold. Second, we prove a pinching theorem for closed minimal submanifolds in a complete simply connected pinched Riemannian manifold, which generalizes the results obtained by S. S. Chern, M. do Carmo, and S. Kobayashi and A. M. Li and J. M. Li respectively. Finally, we obtain a distribution theorem for the square norm of the second fundamental form of M M under the assumption that M M is a minimal submanifold with parallel second fundamental form in a Riemannian manifold.

The curvature of submanifolds of anS-space form

Many authors have studied the geometry of submanifolds of Kaehlerian and Sasakian manifolds. On the other hand, David E. Blair has initiated the study of S-manifolds, which reduce, in particular cases, to Sasakian manifolds [1]. I. Mihai [7] and Ornea [8] have studied CR-submanifolds of S-manifolds. The purpose of the present paper is to investigate some properties ofinvariant and anti-invariant submanifolds of an S-manifold whose invariant f-sectional curvature is constant, that is, of an S-space form. Specifically, those ones related with the curvature tensor fields and with the scalar curvature on the submanifold. In Section 1 we review basic formulas for submanifolds in Riemannian manifolds and, in Section 2, for S-manifolds. In Sections 3 and 4 we study anti-invariant and invariant submanifolds, respectively, of an S-space form. Finally, in the last section we give some examples.

On the instability of minimal submanifolds in Riemannian manifolds of positive curvature

Pour g≥1 et n≥3, il existe une constante δ(g,n) (resp. δ'(g,n))>1/4 telle que si M n est une variete de Riemann δ-pincee simplement connexe compacte a n dimension avec δ>δ(g,n) (resp. δ'(g,n)), alors une immersion minimale ∑ g →M d'une surface de Riemann fermee orientable (resp. non-orientable) ∑ g de genre g est instable

A note on minimal submanifolds in Riemannian manifolds

A note on minimal submanifolds in Riemannian manifolds