Integral Curvature Research Articles

Sharp upper diameter bounds for compact shrinking Ricci solitons

We give a sharp upper diameter bound for a compact shrinking Ricci soliton in terms of its scalar curvature integral and the Perelman’s entropy functional. The sharp cases could occur at round spheres. The proof mainly relies on a sharp logarithmic Sobolev inequality of gradient shrinking Ricci solitons and a Vitali-type covering argument.

Geometric flux formula for the gravitational Wilson loop

Finding diffeomorphism-invariant observables to characterize the properties of gravity and spacetime at the Planck scale is essential for making progress in quantum gravity. The holonomy and Wilson loop of the Levi-Civita connection are potentially interesting ingredients in the construction of quantum curvature observables. Motivated by recent developments in nonperturbative quantum gravity, we establish new relations in three and four dimensions between the holonomy of a finite loop and certain curvature integrals over the surface spanned by the loop. They are much simpler than a gravitational version of the nonabelian Stokes’ theorem, but require the presence of totally geodesic surfaces in the manifold, which follows from the existence of suitable Killing vectors. We show that the relations are invariant under smooth surface deformations, due to the presence of a conserved geometric flux.

The Vanishing Surface Tension Limit of the Muskat Problem

The Muskat problem, in its general setting, concerns the interface evolution between two incompressible fluids of different densities and viscosities in porous media. The interface motion is driven by gravity and capillarity forces, where the latter is due to surface tension. To leading order, both the Muskat problems with and without surface tension effect are scaling invariant in the Sobolev space $H^{1+\frac{d}{2}}(\mathbb{R}^d)$, where $d$ is the dimension of the interface. We prove that for any subcritical data satisfying the Rayleigh-Taylor condition, solutions of the Muskat problem with surface tension $\frak{s}$ converge to the unique solution of the Muskat problem without surface tension locally in time with the rate $\sqrt{\frak{s}}$ when $\frak{s}\to 0$. This allows for initial interfaces that have unbounded or even not locally square integrable curvature. If in addition the initial curvature is square integrable, we obtain the convergence with optimal rate $\frak{s}$.

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.

On Killing Vector Fields on Riemannian Manifolds

We study the influence of a unit Killing vector field on geometry of Riemannian manifolds. For given a unit Killing vector field w on a connected Riemannian manifold (M,g) we show that for each non-constant smooth function f∈C∞(M) there exists a non-zero vector field wf associated with f. In particular, we show that for an eigenfunction f of the Laplace operator on an n-dimensional compact Riemannian manifold (M,g) with an appropriate lower bound on the integral of the Ricci curvature S(wf,wf) gives a characterization of the odd-dimensional unit sphere S2m+1. Also, we show on an n-dimensional compact Riemannian manifold (M,g) that if there exists a positive constant c and non-constant smooth function f that is eigenfunction of the Laplace operator with eigenvalue nc and the unit Killing vector field w satisfying ∇w2≤(n−1)c and Ricci curvature in the direction of the vector field ∇f−w is bounded below by n−1c is necessary and sufficient for (M,g) to be isometric to the sphere S2m+1(c). Finally, we show that the presence of a unit Killing vector field w on an n-dimensional Riemannian manifold (M,g) with sectional curvatures of plane sections containing w equal to 1 forces dimension n to be odd and that the Riemannian manifold (M,g) becomes a K-contact manifold. We also show that if in addition (M,g) is complete and the Ricci operator satisfies Codazzi-type equation, then (M,g) is an Einstein Sasakian manifold.

Ancient solution of mean curvature flow in space forms

In this paper we investigate the rigidity of ancient solutions of the mean curvature flow with arbitrary codimension in space forms. We first prove that under certain sharp asymptotic pointwise curvature pinching condition the ancient solution in a sphere is either a shrinking spherical cap or a totally geodesic sphere. Then we show that under certain pointwise curvature pinching condition the ancient solution in a hyperbolic space is a family of shrinking spheres. We also obtain a rigidity result for ancient solutions in a nonnegatively curved space form under an asymptotic integral curvature pinching condition.

On the magnitude function of domains in Euclidean space

We study Leinster's notion of magnitude for a compact metric space. For a smooth, compact domain $X\subset \mathbb{R}^{2m-1}$, we find geometric significance in the function $\mathcal{M}_X(R) = \mathrm{mag}(R\cdot X)$. The function $\mathcal{M}_X$ extends from the positive half-line to a meromorphic function in the complex plane. Its poles are generalized scattering resonances. In the semiclassical limit $R \to \infty$, $\mathcal{M}_X$ admits an asymptotic expansion. The three leading terms of $\mathcal{M}_X$ at $R=+\infty$ are proportional to the volume, surface area and integral of the mean curvature. In particular, for convex $X$ the leading terms are proportional to the intrinsic volumes, and we obtain an asymptotic variant of the convex magnitude conjecture by Leinster and Willerton, with corrected coefficients.

Intrinsic time gravity, heat kernel regularization, and emergence of Einstein’s theory

The Hamiltonian of intrinsic time gravity is elucidated. The theory describes Schrödinger evolution of our universe with respect to the fractional change of the total spatial volume. Gravitational interactions are introduced by extending Klauder’s momentric variable with similarity transformations, and explicit spatial diffeomorphism invariance is enforced via similarity transformation with exponentials of spatial integrals. In analogy with Yang–Mills theory, a Cotton–York term is obtained from the Chern–Simons functional of the affine connection. The essential difference is the fundamental variable for geometrodynamics is the metric rather than a gauge connection; in the case of Yang–Mills, there is also no analog of the integral of the spatial Ricci scalar curvature. Heat kernel regularization is employed to isolate the divergences of coincidence limits; apart from an additional Cotton–York term, a prescription in which Einstein’s Ricci scalar potential emerges naturally from the positive-definite self-adjoint Hamiltonian of the theory is demonstrated.

The realization of quantum anomalous Hall effect in two dimensional electron gas

The quantum anomalous Hall effect (QAHE), carrying dissipationless chiral edge states, occurs without any magnetic field. Two main strategies were proposed to host QAHE: the magnetic topological insulator thin films and graphene systems. Only the former one was realized in experiment at low temperature. In this paper, by dealing with the two-dimensional electron gas with an anti-dot lattice, a realistic platform is proposed to host the QAHE with both Chern number and . Based on the calculation of the Berry curvature integral and spacial wave function, the topological nature of the QAH edge states is well demonstrated. In the QAH region, the conductance shows quantized plateaus and their values are robust against Anderson disorder. In addition, we have also studied the effects of the size and shape of the anti-dot lattice on QAHE and they provide extra manners to adjust the system parameters. Taking the advantages of the well developed micro-manufacture technique in semiconductors, the proposal is experimentally accessible in micro-scale.

Qualitative properties of bounded subsolutions of nonlinear PDEs

We study decay and compact support properties of positive and bounded solutions of Δpu≥Λ(u) on the exterior of a compact set of a complete manifold with rotational symmetry. In the same setting, we also give a new characterization of stochastic completeness for the p-Laplacian in terms of a global W1,p-regularity of such solutions. One of the tools we use is a nonlinear version of the Feller property which we investigate on general Riemannian manifolds and which we establish under integral Ricci curvature conditions.

Gap theorems for submanifolds in [formula omitted

In this paper, we obtain some gap theorems for complete immersed submanifolds in Hn×R. For this purpose, we first prove the lower bound estimate of the first eigenvalue of submanifolds in a product space satisfying some curvature conditions. Based on this estimate, we get some Bernstein type theorems for submanifolds in Hn(−1)×R under integral curvature pinching conditions.

Weighted Alexandrov–Fenchel inequalities in hyperbolic space and a conjecture of Ge, Wang, and Wu

We consider a conjecture made by Ge, Wang, and Wu regarding weighted Alexandrov–Fenchel inequalities for horospherically convex hypersurfaces in hyperbolic space (a bound, for some physically motivated weight function, of the weighted integral of the k k th mean curvature in terms of the area of the hypersurface). We prove an inequality very similar to the conjectured one. Moreover, when k k is zero and the ambient space has dimension three, we give a counterexample to the conjectured inequality.

Harmonic mean curvature flow and geometric inequalities

We employ the harmonic mean curvature flow of strictly convex closed hypersurfaces in hyperbolic space to prove Alexandrov-Fenchel type inequalities relating quermassintegrals to the total curvature, which is the integral of Gaussian curvature on the hypersurface. The resulting inequality allows us to use the inverse mean curvature flow to prove Alexandrov-Fenchel inequalities between the total curvature and the area for strictly convex hypersurfaces. Finally, we apply the harmonic mean curvature flow to prove a new class of geometric inequalities for h-convex hypersurfaces in hyperbolic space.

Graphene gets bent

Graphene gets bent

Simple, General Criterion for Onset of Disclination Disorder on Curved Surfaces.

Determining the positions of lattice defects on bounded elastic surfaces with Gaussian curvature is a nontrivial task of mechanical energy optimization. We introduce a simple way to predict the onset of disclination disorder from the shape of the surface. The criterion fixes the value of a weighted integral Gaussian curvature to a universal constant and proves accurate across a great variety of shapes. It provides improved understanding of the limitations to crystalline order in many natural and engineering contexts, such as the assembly of viral capsids.

A new local gradient estimate for a nonlinear equation under integral curvature condition on manifolds

A new local gradient estimate for a nonlinear equation under integral curvature condition on manifolds

Rigidity of Minimal Submanifolds in Space Forms

In this paper, we consider the rigidity for an $n(\geq 4)$-dimensional submanfolds $M^n$ with parallel mean curvature in the space form ${\mathbb M}^{n+p}_c$ when the integral Ricci curvature of $M$ has some bound. We prove that, if $c+H^2>0$ and $\|\mathrm{Ric}_{-}^\lambda\|_{n/2}< \epsilon(n,c, \lambda, H)$ for $\lambda$ satisfying $ \frac{n-2}{n-1} (c+H^2) < \lambda \le c+H^2$, then $M$ is the totally umbilical sphere $\mathbb{S}^n(\tfrac{1}{\sqrt{c+H^2}})$. Here $H$ is the norm of the parallel mean curvature of $M$, and $\epsilon(n,c,\lambda, H)$ is a positive constant depending only on $n, c,\lambda$ and $H$. This extends some of the earlier work of [15] from pointwise Ricci curvature lower bound to inetgral Ricci curvature lower bound.

Volume Functional of Compact 4-Manifolds with a Prescribed Boundary Metric

We prove that a critical metric of the volume functional on a 4-dimensional compact manifold with boundary satisfying a second-order vanishing condition on the Weyl tensor must be isometric to a geodesic ball in a simply connected space form $$\mathbb {R}^{4}$$ , $$\mathbb {H}^{4}$$ or $$\mathbb {S}^{4}.$$ Moreover, we provide an integral curvature estimate involving the Yamabe constant for critical metrics of the volume functional, which allows us to get a rigidity result for such critical metrics.

Breather-induced quantised superfluid vortex filaments and their characterisation

We study and characterise the breather-induced quantised superfluid vortex filaments which correspond to the Kuznetsov-Ma breather and super-regular breather excitations developing from localised perturbations. Such vortex filaments, emerging from an otherwise perturbed helical vortex, exhibit intriguing loop structures corresponding to the large amplitude of breathers due to the dual action of bending and twisting of the vortex. The loop induced by the Kuznetsov-Ma breather emerges periodically as time increases, while the loop structure triggered by the super-regular breather—the loop pair—exhibits striking symmetry breaking due to the broken reflection symmetry of the group velocities of the super-regular breather. In particular, we identify explicitly the generation conditions of these loop excitations by introducing a physical quantity—the integral of the relative quadratic curvature—which corresponds to the effective energy of breathers. Despite the nature of nonlinearity, it is demonstrated that this physical quantity shows a linear correlation with the loop size. These results will deepen our understanding of breather-induced vortex filaments and be helpful for controllable ring-like excitations on the vortices.

A sufficient condition for a hypersurface to be isoparametric

Let $M^n$ be a closed Riemannian manifold on which the integral of the scalar curvature is nonnegative. Suppose $\mathfrak{a}$ is a symmetric $(0,2)$ tensor field whose dual $(1,1)$ tensor $\mathcal{A}$ has $n$ distinct eigenvalues, and $\mathrm{tr}(\mathcal{A}^k)$ are constants for $k=1,\ldots, n-1$. We show that all the eigenvalues of $\mathcal{A}$ are constants, generalizing a theorem of de Almeida and Brito [dB90] to higher dimensions. As a consequence, a closed hypersurface $M^n$ in $S^{n+1}$ is isoparametric if one takes $\mathfrak{a}$ above to be the second fundamental form, giving affirmative evidence to Chern's conjecture.