N-dimensional Riemannian Research Articles

Best constants in second-order Sobolev inequalities on compact Riemannian manifolds in the presence of symmetries

Let (M,g) be a smooth compact 3\leq n-dimensional Riemannian manifold, and G a subgroup of the isometry group of (M,g). We establish the best constants in second-order for a Sobolev inequality when the functions are G-invariant.

A splitting theorem for the weighted measure

Let M be an n-dimensional complete noncompact Riemannian manifold, h be a smooth function on M and dμ = eh dV be the weighted measure. In this article, we prove that when the spectrum of the weighted Laplacian \({\triangle_{\mu}}\) has a positive lower bound λ1(M) > 0 and the m(m > n)-dimensional Bakry-Emery curvature is bounded from below by \({-\frac{m-1}{m-2}\lambda_1(M)}\), then M splits isometrically as R × N whenever it has two ends with infinite weighted volume, here N is an (n − 1)-dimensional compact manifold.

Incompatible elasticity and the immersion of non-flat Riemannian manifolds in Euclidean space

We study a geometric problem that originates from theories of nonlinear elasticity: given a non-flat n-dimensional Riemannian manifold with boundary, homeomorphic to a bounded subset of ℝ n , what is the minimum amount of deformation required in order to immerse it in a Euclidean space of the same dimension? The amount of deformation, which in the physical context is an elastic energy, is quantified by an average over a local metric discrepancy. We derive an explicit lower bound for this energy for the case where the scalar curvature of the manifold is non-negative. For n = 2 we generalize the result for surfaces of arbitrary curvature.

ISOMETRIC IMMERSIONS INTO LORENTZIAN PRODUCTS

We give a necessary and sufficient condition for an n-dimensional Riemannian manifold to be isometrically immersed into one of the Lorentzian products 𝕊n× ℝ1or ℍn× ℝ1. This condition is expressed in terms of its first and second fundamental forms, the tangent and normal projections of the vertical vector field. As applications, we give an equivalent condition in a spinorial way and we deduce the existence of a one-parameter family of isometric maximal deformation of a given maximal surface obtained by rotating the shape operator.

The relative isoperimetric inequality on a conformally parabolic manifold with boundary

For an arbitrary noncompact n-dimensional Riemannian manifold with a boundary of conformally parabolic type it is proved that there exists a conformal change of metric such that a relative isoperimetric inequality of the same form as in the closed n-dimensional Euclidean half-space holds on the manifold with the new metric. This isoperimetric inequality is asymptotically sharp.Bibliography: 6 titles.

Bishop-Gromov type inequality on Ricci limit spaces

In this paper, we study limit spaces of a sequence of n-dimensional complete Riemannian manifolds whose Ricci curvatures have definite lower bound. We will give several measure theoretical properties of such limit spaces.

Extension of a theorem of Shi and Tam

In this note, we prove the following generalization of a theorem of Shi and Tam (J Differ Geom 62:79–125, 2002): Let (Ω, g) be an n-dimensional (n ≥ 3) compact Riemannian manifold, spin when n > 7, with non-negative scalar curvature and mean convex boundary. If every boundary component Σi has positive scalar curvature and embeds isometrically as a mean convex star-shaped hypersurface \({{\hat \Sigma}_i \subset \mathbb{R}^n}\), then $$ \int\limits_{\Sigma_i} H d \sigma \le \int\limits_{{\hat \Sigma}_i} \hat{H} d {\hat \sigma} $$ where H is the mean curvature of Σi in (Ω, g), \({\hat{H}}\) is the Euclidean mean curvature of \({{\hat \Sigma}_i}\) in \({\mathbb{R}^n}\), and where d σ and \({d {\hat \sigma}}\) denote the respective volume forms. Moreover, equality holds for some boundary component Σi if, and only if, (Ω, g) is isometric to a domain in \({\mathbb{R}^n}\). In the proof, we make use of a foliation of the exterior of the \({\hat \Sigma_i}\)’s in \({\mathbb{R}^n}\) by the \({\frac{H}{R}}\)-flow studied by Gerhardt (J Differ Geom 32:299–314, 1990) and Urbas (Math Z 205(3):355–372, 1990). We also carefully establish the rigidity statement in low dimensions without the spin assumption that was used in Shi and Tam (J Differ Geom 62:79–125, 2002).

On the uniqueness of heat flow of harmonic maps and hydrodynamic flow of nematic liquid crystals

For any n-dimensional compact Riemannian manifold (M, g) without boundary and another compact Riemannian manifold (N, h), the authors establish the uniqueness of the heat flow of harmonic maps from M to N in the class C([0, T),W 1,n ). For the hydrodynamic flow (u, d) of nematic liquid crystals in dimensions n = 2 or 3, it is shown that the uniqueness holds for the class of weak solutions provided either (i) for n = 2, u ∈ L ∞ L 2 ∩ L 2 H 1 , ▿ P ∈ L 4/3 L 4/3 , and ▿d ∈ L ∞ L 2 ∩ L 2 H 2 ; or (ii) for n = 3, u ∈ L ∞ L 2 ∩ L 2 H 1 ∩ C ([0, T), L n ), P ∈ L /2 L /2 , and ▿d ∈ L 2 L 2 ∩ C ([0, T), L n ). This answers affirmatively the uniqueness question posed by Lin-Lin-Wang. The proofs are very elementary.

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 $${O(\varepsilon |\, {\rm {ln}}\, \varepsilon|)}$$ .

A quasi-linear elliptic equation with critical growth on compact Riemannian manifold without boundary

Let (M, g) be an N-dimensional compact Riemannian manifold without boundary. When m is a positive integer strictly smaller than N ,w e prove that

On some rigidity results of hypersurfaces in a sphere

We study the weak stability index of an immersion ϕ: M → Sn+1 (1) ⊂ Rn+2 of an n-dimensional compact Riemannian manifold. We prove that the weak stability index of a compact hypersurface M with constant scalar curvature in Sn+1 (1), which is not totally umbilical, is greater than or equal to n + 2 if the mean curvature H1 and H3 are constant, and that the equality holds if and only if M is $\smash{S^m(c)\times S^{n-m}(\sqrt{1-c^2})}$. As an application, we show that the weak stability index of an n-dimensional compact hypersurface with constant scalar curvature in Sn+1 (1), which is neither totally umbilical nor a Clifford hypersurface, is greater than or equal to 2n + 4 if the mean curvature H1 and H3 are constant.

Characterizing spheres by conformal vector fields

In this short note we consider an n-dimensional compact Riemannian manifold (M, g) of constant scalar curvature S = n(n − 1)c and show that the presence of a nontrivial conformal vector field ξ on M forces S to be positive. Then we show that an appropriate control on the energy of ξ makes M to be isometric to the n-sphere Sn(c).

Analysis of a General Family of Regularized Navier–Stokes and MHD Models

We consider a general family of regularized Navier-Stokes and Magnetohydrodynamics (MHD) models on n-dimensional smooth compact Riemannian manifolds with or without boundary, with n greater than or equal to 2. This family captures most of the specific regularized models that have been proposed and analyzed in the literature, including the Navier-Stokes equations, the Navier-Stokes-alpha model, the Leray-alpha model, the Modified Leray-alpha model, the Simplified Bardina model, the Navier-Stokes-Voight model, the Navier-Stokes-alpha-like models, and certain MHD models, in addition to representing a larger 3-parameter family of models not previously analyzed. We give a unified analysis of the entire three-parameter family using only abstract mapping properties of the principle dissipation and smoothing operators, and then use specific parameterizations to obtain the sharpest results. We first establish existence and regularity results, and under appropriate assumptions show uniqueness and stability. We then establish results for singular perturbations, including the inviscid and alpha limits. Next we show existence of a global attractor for the general model, and give estimates for its dimension. We finish by establishing some results on determining operators for subfamilies of dissipative and non-dissipative models. In addition to establishing a number of results for all models in this general family, the framework recovers most of the previous results on existence, regularity, uniqueness, stability, attractor existence and dimension, and determining operators for well-known members of this family.

Estimates for eigenvalues of a clamped plate problem on Riemannian manifolds

In this paper we study eigenvalues of a clamped plate problem on a bounded domain in an n-dimensional complete Riemannian manifold. By making use of Nash's theorem and introducing k free constants, we derive a universal bound for eigenvalues, which solves a problem proposed by Wang and Xia [16].

The structure of $${\phi}$$ -stable minimal hypersurfaces in manifolds of nonnegative P-scalar curvature

Suppose (N n , g) is an n-dimensional Riemannian manifold with a given smooth measure m. The P-scalar curvature is defined as $${P(g)=R^m_\infty(g)=R(g)-2\Delta_g{\rm log}\,\phi-|\nabla_g{\rm log}\,\phi|_g^2}$$ , where $${dm=\phi\,dvol(g)}$$ and R(g) is the scalar curvature of (N n , g). In this paper, under a technical assumption on $${\phi}$$ , we prove that $${\phi}$$ -stable minimal oriented hypersurface in the three-dimensional manifold with nonnegative P-scalar curvature must be conformally equivalent to either the complex plane $${\mathbb{C}}$$ or the cylinder $${\mathbb{R}\times\mathbb{S}^1}$$ .

Low-energy effective action in nonperturbative electrodynamics in curved space-time

We study the heat kernel for the Laplace-type partial differential operator acting on smooth sections of a complex spin-tensor bundle over a generic n-dimensional Riemannian manifold. Assuming that the curvature of the U(1) connection (that we call the electromagnetic field) is constant, we compute the first two coefficients of the nonperturbative asymptotic expansion of the heat kernel which are of zero and the first order in Riemannian curvature and of arbitrary order in the electromagnetic field. We apply these results to the study of the effective action in nonperturbative electrodynamics in four dimensions and derive a generalization of the Schwinger’s result for the creation of scalar and spinor particles in electromagnetic field induced by the gravitational field. We discover a new infrared divergence in the imaginary part of the effective action due to the gravitational corrections, which seems to be a new physical effect.

Estimates for eigenvalues on Riemannian manifolds

In this paper, we investigate eigenvalues of the Dirichlet eigenvalue problem of Laplacian on a bounded domain Ω in an n-dimensional complete Riemannian manifold M. When M is an n-dimensional Euclidean space R n , the conjecture of Pólya is well known: the kth eigenvalue λ k of the Dirichlet eigenvalue problem of Laplacian satisfies λ k ⩾ 4 π 2 ( ω n vol Ω ) 2 n k 2 n , for k = 1 , 2 , … . Li and Yau [P. Li, S.T. Yau, On the Schrödinger equation and the eigenvalue problem, Comm. Math. Phys. 88 (1983) 309–318] (cf. Lieb [E. Lieb, The number of bound states of one-body Schrödinger operators and the Weyl problem, in: Proc. Sympos. Pure Math., vol. 36, 1980, pp. 241–252]) have given a partial solution for the conjecture of Pólya, that is, they have proved 1 k ∑ i = 1 k λ i ⩾ n n + 2 4 π 2 ( ω n vol Ω ) 2 n k 2 n , for k = 1 , 2 , … , which is sharp in the sense of average. In this paper, we consider a general setting for complete Riemannian manifolds. We establish an analog of the Li and Yau's inequality for eigenvalues of the Dirichlet eigenvalue problem of Laplacian on a bounded domain in a complete Riemannian manifold. Furthermore, we obtain a universal inequality for eigenvalues of the Dirichlet eigenvalue problem of Laplacian on a bounded domain in a hyperbolic space H n ( − 1 ) . From it, we prove that when the bounded domain Ω tends to H n ( − 1 ) , all eigenvalues tend to ( n − 1 ) 2 4 .

Isometric immersions into $\mathbb {S}^n\times \mathbb {R}$ and $\mathbb {H}^n\times \mathbb {R}$ and applications to minimal surfaces

We give a necessary and sufficient condition for an n-dimensional Riemannian manifold to be isometrically immersed in S n x ℝ or ℍ n × ℝ in terms of its first and second fundamental forms and of the projection of the vertical vector field on its tangent plane. We deduce the existence of a one-parameter family of isometric minimal deformations of a given minimal surface in S 2 × ℝ or ℍ 2 x R, obtained by rotating the shape operator.

Green’s Function for the Hodge Laplacian on Some Classes of Riemannian and Lorentzian Symmetric Spaces

We compute the Green's function for the Hodge Laplacian on the symmetric spaces M\times\Sigma, where M is a simply connected n-dimensional Riemannian or Lorentzian manifold of constant curvature and \Sigma is a simply connected Riemannian surface of constant curvature. Our approach is based on a generalization to the case of differential forms of the method of spherical means and on the use of Riesz distributions on manifolds. The radial part of the Green's function is governed by a fourth order analogue of the Heun equation.

Regularity of Dirac-Harmonic Maps

For any n-dimensional compact spin Riemannian manifold M with a given spin structure and a spinor bundle Σ M, and any compact Riemannian manifold N, we show an ϵ-regularity theorem for weakly Dirac-harmonic maps (ϕ, ψ):M ⊗ Σ M → N ⊗ ϕ * TN. As a consequence, any weakly Dirac-harmonic map is proven to be smooth when n = 2. A weak convergence theorem for approximate Dirac-harmonic maps is established when n = 2. For n ≥ 3, we introduce the notation of stationary Dirac-harmonic maps and obtain a Liouville theorem for stationary Dirac-harmonic maps in . If, in addition, ψ ∈ W 1,p for some , then we obtain an energy monotonicity formula and prove a partial regularity theorem for any such a stationary Dirac-harmonic map.