Sectional Curvature Research Articles

On Saint-Venant Compatibility and Stress Potentials in Manifolds with Boundary and Constant Sectional Curvature

We address three related problems in the theory of elasticity, formulated in the framework of double forms: the Saint-Venant compatibility condition, the existence and uniqueness of solutions for equations arising in incompatible elasticity, and the existence of stress potentials. The scope of this work is for manifolds with boundary of arbitrary dimension, having constant sectional curvature. The central analytical machinery is the regular ellipticity of a boundary-value problem for a bilaplacian operator, and its consequences, which were developed in [R. Kupferman and R. Leder, arXiv:2103.16823, 2021]. One of the novelties of this work is that stress potentials can be used in non-Euclidean geometries, and that the gauge freedom can be exploited to obtain a generalization for the biharmonic equation for the stress potential in dimensions greater than two.

Non-negative divisors and the Grauert metric

Grauert showed that it is possible to construct complete Kähler metrics on the complement of complex analytic sets in a domain of holomorphy. In this note, we study the holomorphic sectional curvatures of such metrics on the complement of a principal divisor in \(\mathbb {C}^n\), \(n \ge 1\). In addition, we also study how this metric and its holomorphic sectional curvature behave when the corresponding principal divisors vary continuously.

Moduli space of non-negative sectional or positive Ricci curvature metrics on sphere bundles over spheres and their quotients

We show that the moduli space of positive Ricci curvature metrics on all the total spaces of S^7-bundles over S^8 which are rational homology spheres has infinitely many path components. Furthermore, we carry out the diffeomorphism classification of quotients of Milnor spheres by a certain involution and show that the moduli space of metrics of non-negative sectional curvature on them has infinitely many path components. Finally, a diffeomorphism finiteness result is obtained on quotients of Shimada spheres by the same type of involution and we show that for the types that can be expressed by an infinite family of manifolds, the moduli space of positive Ricci curvature metrics has infinitely many path components.

Three circle theorem on almost Hermitian manifolds and applications

In this paper, we extend Gang Liu’s three circle theorem for Kähler manifolds to almost Hermitian manifolds. As applications of the three circle theorem, we obtain sharp dimension estimates for spaces of holomorphic functions of polynomial growth and rigidity for the estimates, Liouville theorems for pluri-subharmonic functions of sub-logarithmic growth and Liouville theorems for holomorphic functions of Cheng-type, and fundamental theorem of algebra on almost Hermitian manifolds with slightly negative holomorphic sectional curvature. The results are generalization of Liu’s results and some of them are new even for the Kähler case. We also discuss the converse of the three circle theorem on Hermitian manifolds which turns out to be rather different with the Kähler case. In order to obtain the three circle theorem on almost Hermitian manifolds, we also establish a general maximum principle in the spirit of Calabi’s trick so that a general three circle theorem is a straightforward consequence of the general maximum principle. The general maximum principle and three circle theorem established may be of independent interests.

Topology and Curvature of Isoparametric Families in Spheres

An isoparametric family in the unit sphere consists of parallel isoparametric hypersurfaces and their two focal submanifolds. The present paper has two parts. The first part investigates topology of the isoparametric families, namely the homotopy, homeomorphism, or diffeomorphism types, parallelizability, as well as the Lusternik–Schnirelmann category. This part extends substantially the results of Wang (J Differ Geom 27:55–66, 1988). The second part is concerned with their curvatures; more precisely, we determine when they have non-negative sectional curvatures or positive Ricci curvatures with the induced metric.

Extrinsic eigenvalues upper bounds for submanifolds in weighted manifolds

We prove Reilly-type upper bounds for divergence-type operators of the second order as well as for Steklov problems on submanifolds of Riemannian manifolds of bounded sectional curvature endowed with a weighted measure.

Unbounded Pontryagin numbers on nonnegatively curved spin manifolds

We prove that any rational linear combination of Pontryagin numbers that does not factor through the universal elliptic genus is unbounded on connected closed spin manifolds of nonnegative sectional curvature.

Diffeomorphic souls and disconnected moduli spaces of nonnegatively curved metrics

We give examples of open manifolds that carry infinitely many complete metrics of nonnegative sectional curvature such that they all have the same soul, and their isometry classes lie in different connected components of the moduli space. All previously known examples of this kind have souls of codimension one. In our examples the souls have codimensions three and two.

Rotation hypersurfaces in semi-Riemannian space forms of arbitrary index

Let n, ν be integers such that n≥2 and 0≤ν≤n+1. Let Mc,νn+1 be a (n+1)-dimensional semi-Riemannian space form of constant sectional curvature c and index ν. We give the parametrizations of rotation hypersurfaces in Mc,νn+1. Adding the condition for the r-th mean curvature to be constant, we show the study of these hypersurfaces is reduced to the study of a function on two variables. As an illustrative example, we study the particular case of rotation hypersurfaces with constant scalar curvature in Mc,νn+1, determining the values of n,c,ν for which these hypersurfaces are isoparametric. In the final part we study the variation of the second fundamental form of rotation hypersurfaces in order to obtain some rigidity results.

Characterization of Whitney spheres among Lagrangian submanifolds with conformal Maslov form

In this paper, we give new characterizations of Whitney spheres among n-dimensional compact and non-minimal Lagrangian submanifolds immersed in the complex space form Nn(4c) with constant holomorphic sectional curvature 4c. We first show that an n-dimensional compact and non-minimal Lagrangian submanifolds satisfying a point-wise pinching condition in Nn(4c) is a Whitney sphere. Then we prove that there is a positive constant C(n) depending only on n such that, for a compact and non-minimal submanifold Mn immersed in Nn(4c) with c≥0, if the modified second fundamental form B satisfies ∫M|B|ndμ<C(n), then B≡0 and Mn is a Whitney sphere.

Golden Riemannian Manifolds Having Constant Sectional Curvatures and Their Submanifolds

Golden Riemannian Manifolds Having Constant Sectional Curvatures and Their Submanifolds

Fast seismic response estimation of tall pier bridges based on deep learning techniques

Seismic responses of tall pier bridges are usually estimated with nonlinear time history analysis (NLTHA) since it is able to provide rigorous results while the time consumption is acceptable with the improvement of computers. Note that parallel computing employing multiple computers might be required to facilitate estimating the performance of numerous bridges in highway networks after earthquakes. Recently, deep learning techniques have been recognized as promising alternatives for predicting structural responses in earthquake engineering with significantly improved time efficiency. Therefore, this paper develops a fast seismic performance estimation methodology using deep learning procedures to rapidly predict the seismic demands of tall pier bridges. The efficiency of the employed techniques is verified through illustrative examples, by comparing the predicted responses with those obtained from NLTHA under several types of input motions. The results show that when trained following appropriate steps, the deep learning models could provide satisfactory prediction for shear force, bending moment, as well as section curvature ductility. Additionally, the time efficiency of deep learning models is shown increased by about 97% compared with NLTHA, which might be further improved for more complex structural systems. Further parametric analysis reveals that the efficiency of selecting proper input variables for deep learning models could be significantly improved by considering the physical characteristics of structures; e.g., structural dynamic properties and interaction between structure and ground motion. This methodology is believed especially favored evaluating the seismic performance/post-earthquake resilience of highway networks containing thousands of bridges, in which conducting NLTHA for each bridge is prohibitively computational demanding and might delay rescue operations.

Ricci-like Solitons with Arbitrary Potential and Gradient Almost Ricci-like Solitons on Sasaki-like Almost Contact B-metric Manifolds

Ricci-like solitons with arbitrary potential are introduced and studied on Sasaki-like almost contact B-metric manifolds. A manifold of this type can be considered as an almost contact complex Riemannian manifold which complex cone is a holomorphic complex Riemannian manifold. The soliton under study is characterized and proved that its Ricci tensor is equal to the vertical component of both B-metrics multiplied by a constant. Thus, the scalar curvatures with respect to both B-metrics are equal and constant. In the 3-dimensional case, it is found that the special sectional curvatures with respect to the structure are constant. Gradient almost Ricci-like solitons on Sasaki-like almost contact B-metric manifolds have been proved to have constant soliton coefficients. Explicit examples are provided of Lie groups as manifolds of dimensions 3 and 5 equipped with the structures under study.

Vector bundles of non-negative curvature over cohomogeneity one manifolds

We provide several results on the existence of metrics of non-negative sectional curvature on vector bundles over certain cohomogeneity one manifolds and homogeneous spaces up to suitable stabilization.Beside explicit constructions of the metrics, this is achieved by identifying equivariant structures upon these vector bundles via a comparison of their equivariant and non-equivariant K-theory. For this, in particular, we transcribe equivariant K-theory to equivariant rational cohomology and investigate surjectivity properties of induced maps in the Borel fibration via rational homotopy theory.

Sobolev Inequalities in Manifolds with Nonnegative Curvature

AbstractWe prove a sharp Sobolev inequality on manifolds with nonnegative Ricci curvature. Moreover, we prove a Michael‐Simon inequality for submanifolds in manifolds with nonnegative sectional curvature. Both inequalities depend on the asymptotic volume ratio of the ambient manifold. © 2022 Wiley Periodicals LLC.

Pluriclosed Manifolds with Constant Holomorphic Sectional Curvature

A long-standing conjecture in complex geometry says that a compact Hermitian manifold with constant holomorphic sectional curvature must be Kähler when the constant is non-zero and must be Chern flat when the constant is zero. The conjecture is known in complex dimension 2 by the work of Balas-Gauduchon in 1985 (when the constant is zero or negative) and by Apostolov—Davidov—Muskarov in 1996 (when the constant is positive). For higher dimensions, the conjecture is still largely unknown. In this article, we restrict ourselves to pluriclosed manifolds, and confirm the conjecture for the special case of Strominger Kähler-like manifolds, namely, for Hermitian manifolds whose Strominger connection (also known as Bismut connection) obeys all the Kähler symmetries.

Some results on Kenmotsu statistical manifolds

In this paper, we first investigate the Kenmotsu statistical structures built on a Kenmotsu space form and determine some special Kenmotsu statistical structures under two curvature conditions. Secondly, we show that if the holomorphic sectional curvature of the hypersurface orthogonal to the structure vector in a Kenmotsu statistical manifold is constant, then the $\phi-$sectional curvature of the ambient Kenmotsu statistical manifold must be constant $-1$, and the constant holomorphic sectional curvature of the hypersurface is $0$. In addition, some non-trivial examples are given to illustrate the results of this paper.

Counting minimal surfaces in negatively curved 3-manifolds

We introduce an asymptotic quantity that counts area-minimizing surfaces in negatively curved closed 3-manifolds and show that quantity to only be minimized, among all metrics of sectional curvature ≤−1, by the hyperbolic metric.

On projective manifolds with semi-positive holomorphic sectional curvature

In this paper, we establish a structure theorem for a smooth projective variety $X$ with semi-positive holomorphic sectional curvature. Our structure theorem contains the solution for Yau's conjecture and it can be regarded as a natural generalization of the structure theorem proved by Howard-Smyth-Wu and Mok for holomorphic bisectional curvature. Specifically, we prove that $X$ admits a locally trivial morphism $\phi:X \to Y$ such that the fiber $F$ is rationally connected and the image $Y$ has a finite \'etale cover $A \to Y$ by an abelian variety $A$, by combining the author's previous work with the theory of holomorphic foliations. Moreover, we show that the universal cover of $X$ is biholomorphic and isometric to the product $\mathbb{C}^m \times F$ of the universal cover $\mathbb{C}^m$ of $Y$ with a flat metric and the rationally connected fiber $F$ with a K\"ahler metric whose holomorphic sectional curvature is semi-positive.

Asymptotic rigidity for shells in non-Euclidean elasticity

We consider a prototypical “stretching plus bending” functional of an elastic shell. The shell is modeled as a d-dimensional Riemannian manifold endowed, in addition to the metric, with a reference second fundamental form. The shell is immersed into a (d+1)-dimensional ambient space, and the elastic energy accounts for deviations of the induced metric and second fundamental forms from their reference values. Under the assumption that the ambient space is of constant sectional curvature, we prove that any sequence of immersions of asymptotically vanishing energy converges to an isometric immersion of the shell into ambient space, having the reference second fundamental form. In particular, if the ambient space is Euclidean space, then the reference metric and second fundamental form satisfy the Gauss-Codazzi-Mainardi compatibility conditions. This theorem can be viewed as a (manifold-valued) co-dimension 1 analog of Reshetnyak's asymptotic rigidity theorem. It also relates to recent results on the continuity of surfaces with respect to their fundamental forms.