Curvature Conditions Research Articles

Lichnerowicz-type estimates for the first eigenvalue of biharmonic operator

In this paper we are going to investigate the Lichnerowicz-type estimate theorem for the first eigenvalue of buckling and clamped plate problems on both Riemannian and Kähler manifolds. Also we will study both problems in a case of the integral curvature condition on the Kähler manifold.

Compact null hypersurfaces in Lorentzian manifolds

Abstract We show how the topological and geometric properties of the family of null hypersurfaces in a Lorentzian manifold are related with the properties of the ambient manifold itself. In particular, we focus in how the presence of global symmetries and curvature conditions restrict the existence of compact null hypersurfaces. We use these results to show the influence on the existence of compact totally umbilic null hypersurfaceswhich are not totally geodesic. Finally we describe the restrictions that they impose in causality theory.

Improved generalized periods estimates over curves on Riemannian surfaces with nonpositive curvature

Abstract We show that, on compact Riemannian surfaces of nonpositive curvature, the generalized periods, i.e. the 𝜈-th order Fourier coefficients of eigenfunctions e λ e_{\lambda} over a closed smooth curve 𝛾 which satisfies a natural curvature condition, go to 0 at the rate of O ⁢ ( ( log ⁡ λ ) - 1 2 ) O((\log\lambda)^{-\frac{1}{2}}) in the high energy limit λ → ∞ \lambda\to\infty if 0 < | ν | λ < 1 - δ 0<\frac{\lvert\nu\rvert}{\lambda}<1-\delta for any fixed 0 < δ < 1 0<\delta<1 . Our result implies, for instance, that the generalized periods over geodesic circles on any surfaces with nonpositive curvature would converge to zero at the rate of O ⁢ ( ( log ⁡ λ ) - 1 2 ) O((\log\lambda)^{-\frac{1}{2}}) .

Positivity of Curvature On Manifolds With Boundary

Abstract Consider a compact manifold $M$ with smooth boundary $\partial M$. Suppose that $g$ and $\tilde{g}$ are two Riemannian metrics on $M$. We construct a family of metrics on $M$ that agrees with $g$ outside a neighborhood of $\partial M$ and agrees with $\tilde{g}$ in a neighborhood of $\partial M$. We prove that the family of metrics preserves various natural curvature conditions under suitable assumptions on the boundary data. Moreover, under suitable assumptions on the boundary data, we can deform a metric to one with totally geodesic boundary while preserving various natural curvature conditions.

Volume preserving flow and Alexandrov–Fenchel type inequalities in hyperbolic space

In this paper, we study flows of hypersurfaces in hyperbolic space, and apply them to prove geometric inequalities. In the first part of the paper, we consider volume preserving flows by a family of curvature functions including positive powers of k -th mean curvatures with k=1,\ldots,n , and positive powers of p -th power sums S_p with p > 0 . We prove that if the initial hypersurface M_0 is smooth and closed and has positive sectional curvatures, then the solution Mt of the flow has positive sectional curvature for any time t > 0 , exists for all time and converges to a geodesic sphere exponentially in the smooth topology. The convergence result can be used to show that certain Alexandrov–Fenchel quermassintegral inequalities, known previously for horospherically convex hypersurfaces, also hold under the weaker condition of positive sectional curvature. In the second part of this paper, we study curvature flows for strictly horospherically convex hypersurfaces in hyperbolic space with speed given by a smooth, symmetric, increasing and degree one homogeneous function f of the shifted principal curvatures \lambda_i=\kappa_i-1 , plus a global term chosen to impose a constraint on the quermassintegrals of the enclosed domain, where f is assumed to satisfy a certain condition on the second derivatives. We prove that if the initial hypersurface is smooth, closed and strictly horospherically convex, then the solution of the flow exists for all time and converges to a geodesic sphere exponentially in the smooth topology. As applications of the convergence result, we prove a new rigidity theorem on smooth closed Weingarten hypersurfaces in hyperbolic space, and a new class of Alexandrov–Fenchel type inequalities for smooth horospherically convex hypersurfaces in hyperbolic space.

On the structure of Ricci shrinkers

We develop a structure theory for non-collapsed Ricci shrinkers without any curvature condition. As applications, we obtain some curvature estimates of the Ricci shrinkers depending only on the non-collapsing constant.

A formula for the time derivative of the entropic cost and applications

In the recent years the Schrödinger problem has gained a lot of attention because of the connection, in the small-noise regime, with the Monge-Kantorovich optimal transport problem. Its optimal value, the entropic costCT, is here deeply investigated. In this paper we study the regularity of CT with respect to the parameter T under a curvature condition and explicitly compute its first and second derivative. As applications:-we determine the large-time limit of CT and provide sharp exponential convergence rates; we obtain this result not only for the classical Schrödinger problem but also for the recently introduced Mean Field Schrödinger problem [3];-we improve the Taylor expansion of T↦TCT around T=0 from the first to the second order.

Rigidity of a Thin Domain Depends on the Curvature, Width, and Boundary Conditions

This paper is concerned with the study of linear geometric rigidity of shallow thin domains under zero Dirichlet boundary conditions on the displacement field on the thin edge of the domain. A ribbon is a thin domain that has in-plane dimensions of order O(1) and \(\epsilon ,\) where \(\epsilon \in (h,1)\) is a parameter (here h is the thickness of the domains). The problem has been solved by Grabovsky and the second author in (Ann de l’Inst Henri Poincare (C) An Non Lin 2018 35(1):267–282, 2018) and by the second author in (Arch Ration Mech Anal 226(2):743–766, 2017) for the case \(\epsilon =1,\) with the outcome of the optimal constant \(C\sim h^{-3/2},\) \(C\sim h^{-4/3},\) and \(C\sim h^{-1}\) for parabolic, hyperbolic, and elliptic thin domains respectively. We prove in the present work that in fact there are two distinctive scaling regimes \(\epsilon \in (h,\sqrt{h}]\) and \(\epsilon \in (\sqrt{h},1),\) such that in each of which the thin domain rigidity is given by a certain formula in h and \(\epsilon .\) An interesting new phenomenon is that in the first (small parameter) regime \(\epsilon \in (h,\sqrt{h}]\), the rigidity does not depend on the curvature of the thin domain mid-surface.

SOME NOTES ON KENMOTSU MANIFOLD

In the present paper, we deal with a Kenmotsu manifold $M$. Firstly, we study the notion of torse-forming vector field on such a manifold. Then, we investigate some curvature conditions such as $Q.\mathcal{M}=0$ and $C.Q=0$ on such a manifold and obtain some necessary conditions for such a manifold given as to be Einstein and $\eta-$Einstein. Also, we study a Kenmotsu manifold $M$ admitting a Ricci soliton and give an example for this manifold.

Chiti-type Reverse Hölder Inequality and Torsional Rigidity Under Integral Ricci Curvature Condition

In this paper, we prove a reverse Holder inequality for the eigenfunction of the Dirichlet problem on domains of a compact Riemannian manifold with the integral Ricci curvature condition. We also prove an isoperimetric inequality for the torsional rigidity of such domains. These results extend some recent work of Gamara et al. (Open Math. 13(1), 557–570, 2015) and Colladay et al. (J. Geom. Anal. 28(4), 3906–3927, 2018) from the pointwise lower Ricci curvature bound to the integral Ricci curvature condition. We also extend the results from Laplacian to p-Laplacian.

D-homothetic deformation of (κ,µ) manifold

In this paper, we study the invariance of certain curvature conditions in (κ,µ)-contact metric manifold under D-homothetic deformation. Finally we give an example to verify the results.

A note on concircular curvature tensor in Lorentzian almost para-contact geometry

The paper deals with the notion of different classes of concircular curvature tensor on Lorentzian almost para-contact manifolds admitting a quarter-symmetric metric connection. In this paper we study Lorentzian almost para-contact manifolds with respect to the quarter-symmetric metric connection satisfying the curvature condition Z.S = 0. We also investigate the properties of ξ−concircularly flat, φ−concircularly flat and quasi-concircularly flat Lorentzian almost para-contact manifolds admitting a quarter-symmetric metric connection and it is found that in each of above cases the manifold is generalized η−Einstein manifold.

On a type of quarter-symmetric non-recurrent metric connection on a p-Sasakian manifold

We study a Para-Sasakian manifold admitting a type of quartersymmetric non-recurrent-metric connection whose concircular curvature tensor satisfies certain curvature conditions.

Shadows of graphical mean curvature flow

We consider mean curvature flow of an initial surface that is the graph of a function over some domain of definition in $R^n$. If the graph is not complete then we impose a constant Dirichlet boundary condition at the boundary of the surface. We establish longtime-existence of the flow and investigate the projection of the flowing surface onto $R^n$, the shadow of the flow. This moving shadow can be seen as a weak solution for mean curvature flow of hypersurfaces in $R^n$ with a Dirichlet boundary condition. Furthermore, we provide a lemma of independent interest to locally mollify the boundary of an intersection of two smooth open sets in a way that respects curvature conditions.

On hypersurfaces satisfying conditions determined by the Opozda–Verstraelen affine curvature tensor

Using the Blaschke-Berwald metric and the affine shape operator of a hypersurface M in the (n+1)-dimensional real affine space we can define some generalized curvature tensor named the Opozda-Verstraelen affine curvature tensor. In this paper we determine curvature conditions of pseudosymmetry type expressed by this tensor for locally strongly convex hypersurfaces M, n>2, with two distinct affine principal curvatures or with three distinct affine principal curvatures assuming that at least one affine principal curvature has multiplicity 1.

A safety assessment for the Guanlei Port – Qingsheng Port water corridors on the Lancang – Mekong River

ABSTRACT The water corridors between the Guanlei Port and the Qingsheng Port on the Lancang – Mekong River are important waterway for Yunnan and Southeast Asian countries. However, there are certain unpredictable safety hazards due to its location bordering multiple countries, complex terrain, sparse human settlements and inconvenient transportation. To help enhancing the safety of the Lancang – Mekong River international waterway, this paper used satellite imagery to interpret safety factors such as fault structures, channel width, rapids, vegetation coverage, river curvature, human settlements and transportation conditions. Based on analytic hierarchy process (AHP) , this research assessed the shipping safety of the waterway, analysed the effects of various factors on navigation, and made suggestions on the shipping safety which is important in the border areas. The results show that the Meisai section of the Mekong River has the shipping safety at level 1 where the limited joint patrol is proposed. The navigation channel from Guanlei Port to Wanxianwotang has the shipping safety at levels 2 & 3 where the intermediate effort of the joint patrol is suggested. The waterway from Wanxianwotang to Wannanpen has the shipping safety at levels 4 & 5 where the strong presence of the joint patrol is recommended.

The fundamental group, rational connectedness and the positivity of Kähler manifolds

Abstract Firstly, we confirm a conjecture asserting that any compact Kähler manifold N with Ric ⊥ > 0 {\operatorname{Ric}^{\perp}>0} must be simply-connected by applying a new viscosity consideration to Whitney’s comass of ( p , 0 ) {(p,0)} -forms. Secondly we prove the projectivity and the rational connectedness of a Kähler manifold of complex dimension n under the condition Ric k > 0 {\operatorname{Ric}_{k}>0} (for some k ∈ { 1 , … , n } {k\in\{1,\dots,n\}} , with Ric n {\operatorname{Ric}_{n}} being the Ricci curvature), generalizing a well-known result of Campana, and independently of Kollár, Miyaoka and Mori, for the Fano manifolds. The proof utilizes both the above comass consideration and a second variation consideration of [L. Ni and F. Zheng, Positivity and Kodaira embedding theorem, preprint 2020, https://arxiv.org/abs/1804.09696]. Thirdly, motivated by Ric ⊥ {\operatorname{Ric}^{\perp}} and the classical work of Calabi and Vesentini [E. Calabi and E. Vesentini, On compact, locally symmetric Kähler manifolds, Ann. of Math. (2) 71 1960, 472–507], we propose two new curvature notions. The cohomology vanishing H q ⁢ ( N , T ′ ⁢ N ) = { 0 } {H^{q}(N,T^{\prime}N)=\{0\}} for any 1 ≤ q ≤ n {1\leq q\leq n} and a deformation rigidity result are obtained under these new curvature conditions. In particular, they are verified for all classical Kähler C-spaces with b 2 = 1 {b_{2}=1} . The new conditions provide viable candidates for a curvature characterization of homogeneous Kähler manifolds related to a generalized Hartshone conjecture.

GAUGE EQUIVALENCE BETWEEN THE TWO-COMPONENT GENERALIZATION OF THE (2+1)-DIMENSIONAL DAVEY-STEWARTSON I EQUATION AND 𝜞 - SPIN SYSTEM

In recent years, multidimensional nonlinear evolutionary equations have been actively studied within the framework of the theory of solitons. Their relevance is confirmed by numerous scientific publications. In this work the gauge equivalence between the (2+1)-dimensional integrable two-component Davey-Stewartson I (DSI) equation and the Г - spin system is established. Multicomponent generalizations of nonlinear integrable equations are of current interest from both physical and mathematical points of view. In the theory of integrable (soliton) equations, one of the key models is integrable nonlinear Schrodinger-type (NLS) equations. A two-component integrable generalization of the (2+1)-dimensional DSI equation, obtained on the basis of its one-component representation, and its corresponding Lax representation were proposed. A geometric connection between the twolayer spin system and the integrable two-component Manakov system is found. The nonlinear equations are integrated using the inverse scattering problem method by means of a linear system. For each integrable nonlinear equation, as is known, there is a Lax pair of two linear equations, a compatibility condition, that is, a condition of zero curvature, which this equation serves. We have obtained Lax pair whose zero curvature condition gives the Г - spin system.

Projective Curvature Tensor on N($\kappa$)-Contact Metric Manifold Admitting Semi-Symmetric Non-Metric Connection

The object of the present paper is to classify $N(\kappa)$-contact metric manifolds admitting the semi-symmetric non-metric connection with certain curvature conditions the projectively curvature tensor. We studied projective flat, $\xi- $projectively flat, $\phi- $projectively flat $N(\kappa )$-contact metric manifolds admitting the semi-symmetric non-metric connection. Also, we examine such manifolds under some local symmetry conditions related to projective curvature tensor.

Lp bounds of maximal operators along variable planar curves in the Lipschitz regularity

In this paper, for general plane curves γ satisfying some suitable smoothness and curvature conditions, we obtain the single annulus Lp(R2)-boundedness of the Hilbert transforms HU,γ∞ along the variable plane curves (t,U(x1,x2)γ(t)) and the Lp(R2)-boundedness of the corresponding maximal functions MU,γ∞, where p>2 and U is a measurable function. The range on p is sharp. Furthermore, for 1<p≤2, under the additional conditions that U is Lipschitz and making a ε0-truncation with γ(2ε0)≤1/4‖U‖Lip, we also obtain similar boundedness for these two operators HU,γε0 and MU,γε0.