N-dimensional Compact Riemannian Manifold Research Articles

A new approach from the calculus of variations to the Lichnerowicz theorem

The well-known Lichnerowicz theorem states that on an n-dimensional compact Riemannian manifold without boundary with the assumptionRicg≥(n−1)g on the Ricci curvature, the first eigenvalue λ of the Laplacian on the manifold satisfiesλ≥n. This theorem is proved using a Bochner formula.In this paper we give a new approach from the calculus of variations to the Lichnerowicz theorem. We use the energy Econf() of conformality, introduced in [3]. For a smooth map f between Riemannian manifolds, the map f is a weakly conformal map (almost everywhere) if and only if Econf(f) = 0 (the minimum value).Our approach is as follows:(1)The identity map on M is a conformal map, hence a minimizer of the energy Econf(). Especially the identity map is stable, i.e., the second variation is non-negative for the identity map.(2)For the first eigenfunction φ, we take the variation vector field corresponding to dφ, and the non-negativity of the second variation for the variation vector field at the identity map implies the Lichnerowicz theorem.

Density of the Level Sets of the Metric Mean Dimension for Homeomorphisms

AbstractLet N be an n-dimensional compact riemannian manifold, with $$n\ge 2$$ n ≥ 2 . In this paper, we prove that for any $$\alpha \in [0,n]$$ α ∈ [ 0 , n ] , the set consisting of homeomorphisms on N with lower and upper metric mean dimensions equal to $$\alpha $$ α is dense in $$\text {Hom}(N)$$ Hom ( N ) . More generally, given $$\alpha ,\beta \in [0,n]$$ α , β ∈ [ 0 , n ] , with $$\alpha \le \beta $$ α ≤ β , we show the set consisting of homeomorphisms on N with lower metric mean dimension equal to $$\alpha $$ α and upper metric mean dimension equal to $$\beta $$ β is dense in $$\text {Hom}(N)$$ Hom ( N ) . Furthermore, we also give a proof that the set of homeomorphisms with upper metric mean dimension equal to n is residual in $$\text {Hom}(N)$$ Hom ( N ) .

Maximal estimates for an oscillatory operator

Let M be an n-dimensional compact Riemannian manifold. We study an oscillatory operator Tα,β,t and obtain its maximal estimate on Lp. Particularly, we obtain the sharp estimate on L1 case. As an application, we obtain the convergence rate of the fractional Schrödinger propagator on the Sobolev space.

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.

Maclaurin Heat Coefficients and Associated Zeta Functions on Quaternionic Projective Spaces Pn(H) (n ≥ 1)

The heat invariants or the Minakshisundaram-Pleijel heat coefficients (M) (k ≥ 0) describe the asymptotic expansion of the heat kernel HM on any N = 4 n-dimensional (n ≥ 1) compact Riemannian manifold M; associated with the coefficients is the Minakshisundaram-Pleijel zeta function ζM = ζM(s) (s ∈ C). In this paper, we introduce and study a new class of heat coefficients, namely, the Maclaurin heat coefficients (t > 0 , m 0) (i.e., the coefficients appearing in the Maclaurin expansion of the heat kernel HM(t, θ)) in terms of the classical and generalised Minakshisundaram-Pleijel coefficients and (M) (0 ⩽ j ⩽ m) respectively, when M = P n(H) (n ≥ 1), a quaternionic projective space. Remarkable asymptotic expansions for the Maclaurin spectral functions are established. We also introduce and construct new zeta functions (m ≥ 0) associated with these Maclaurin heat coefficients (generalised Minakshisundaram-Pleijel zeta functions), and it is interesting to see that these generalised zeta functions can be explicitly understood in terms of the classical (Minakshisundaram-Pleijel) zeta functions.

On Compact Riemannian Manifolds with Harmonic Weyl Curvature

We give some rigidity theorems for an n-dimensional ( $$n\ge 4$$ ) compact Riemannian manifold with harmonic Weyl curvature, positive scalar curvature and positive constant $$\sigma _2$$ curvature. Moreover, we prove that a 4-dimensional compact locally conformally flat Riemannian manifold with positive scalar curvature and positive constant $$\sigma _2$$ curvature is isometric to a quotient of the round $$\mathbb {S}^4$$ .

Elliptic equation with critical and negative exponents involving the GJMS operator on compact Riemannian manifolds

In this paper we consider an elliptic equation with critical and negative exponents involving the GJMS (Graham–Jenne–Mason–Sparling) operator of order k on n-dimensional compact Riemannian manifolds with 2k<n. Multiplicity and nonexistence results are established.

Variation of the first eigenvalue of p-Laplacian on evolving geometry and applications

Let (M,g)be an n-dimensional compact Riemannian manifold whose metric g(t)evolves by the generalised abstractgeometric flow. This paper discusses the variation formulas, monotonicity and differentiability for the first eigenvalue of thep-Laplacian on (M,g(t)). It is shown that the first nonzero eigenvalue is monotonically nondecreasing along the flow undercertain geometric conditions and that it is differentiable almost everywhere. These results provide a unified approach to the study of eigenvalue variations and applications under many geometric flows

Eigenvalues of the weighted Laplacian under the extended Ricci flow

Abstract Let ∆φ = ∆ − ∇φ∇ be a symmetric diffusion operator with an invariant weighted volume measure dμ = e −φ dν on an n-dimensional compact Riemannian manifold (M, g), where g = g(t) solves the extended Ricci flow. We study the evolution and monotonicity of the first nonzero eigenvalue of ∆φ and we obtain several monotone quantities along the extended Ricci flow and its volume preserving version under some technical assumption. We also show that the eigenvalues diverge in a finite time for n ≥ 3. Our results are natural extensions of some known results for Laplace–Beltrami operators under various geometric flows.

Looping Directions and Integrals of Eigenfunctions over Submanifolds

Let $$(M,\,g)$$ be a compact n-dimensional Riemannian manifold without boundary and $$e_\lambda $$ be an $$L^2$$ -normalized eigenfunction of the Laplace–Beltrami operator with respect to the metric g, i.e., $$\begin{aligned} -\Delta _g e_\lambda = \lambda ^2 e_\lambda \quad \text {and} \quad \left\| e_\lambda \right\| _{L^2(M)} = 1. \end{aligned}$$ Let $$\varSigma $$ be a d-dimensional submanifold and $$\mathrm{d}\mu $$ a smooth, compactly supported measure on $$\varSigma .$$ It is well known (e.g., proved by Zelditch, Commun Partial Differ Equ 17(1–2):221–260, 1992 in far greater generality) that $$\begin{aligned} \int _\varSigma e_\lambda \, \mathrm{d}\mu = O\left( \lambda ^\frac{n-d-1}{2}\right) . \end{aligned}$$ We show this bound improves to $$o\left( \lambda ^\frac{n-d-1}{2}\right) $$ provided the set of looping directions, $$\begin{aligned} {{\mathcal {L}}}_{\varSigma } = \{ (x,\,\xi ) \in \mathrm{SN}^*\varSigma : \varPhi _t(x,\,\xi ) \in \mathrm{SN}^*\varSigma \text { for some } t > 0 \} \end{aligned}$$ has measure zero as a subset of $$\mathrm{SN}^*\varSigma ,$$ where here $$\varPhi _t$$ is the geodesic flow on the cosphere bundle $$S^*M$$ and $$\mathrm{SN}^*\varSigma $$ is the unit conormal bundle over $$\varSigma .$$

Hypercontractivity Property and General Optimal $$L^p$$ L p -Entropy Inequalities on Riemannian Manifolds

Let (M, g) be a n-dimensional smooth compact Riemannian manifold without boundary with $$n\ge 2$$ . We prove that the optimal Riemannian p-entropy inequality $$\begin{aligned} \int _M |u|^p\log (|u|^p) \; dv_g\le \dfrac{n}{\tau }\log \left[ {\mathcal {A}}(p,\tau )\left( \int _M |\nabla _g u|^p\; dv_g\right) ^{\frac{\tau }{p}}+{\mathcal {B}}(p)\right] \end{aligned}$$ is valid for every $$u\in H^{1,p}(M)$$ with $$\Vert u\Vert _p=1$$ where $$p>1$$ and $$1\le \tau < \min \{2,p\}$$ or $$\tau = p \le 2$$ . Also, we investigated the relationship between this optimal inequality and the hypercontractivity property for the non-linear evolution equation $$\begin{aligned} u_t=\Delta _p\left( u^{\frac{1}{p-1}}\right) ,\quad x\in M,\ t>0. \end{aligned}$$ When $$\tau =p\le 2$$ we find explicit estimates of the time asymptotic behavior of their solutions with initial data in the spaces $$L^q(M)$$ , $$1\le q<\infty $$ .

Large conformal metrics with prescribed scalar curvature

Let (M,g) be an n-dimensional compact Riemannian manifold. Let h be a smooth function on M and assume that it has a critical point ξ∈M such that h(ξ)=0 and which satisfies a suitable flatness assumption. We are interested in finding conformal metrics gλ=uλ4n−2g, with u>0, whose scalar curvature is the prescribed function hλ:=λ2+h, where λ is a small parameter.In the positive case, i.e. when the scalar curvature Rg is strictly positive, we find a family of “bubbling” metrics gλ, where uλ blows up at the point ξ and approaches zero far from ξ as λ goes to zero.In the general case, if in addition we assume that there exists a non-degenerate conformal metric g0=u04n−2g, with u0>0, whose scalar curvature is equal to h, then there exists a bounded family of conformal metrics g0,λ=u0,λ4n−2g, with u0,λ>0, which satisfies u0,λ→u0 uniformly as λ→0. Here, we build a second family of “bubbling” metrics gλ, where uλ blows up at the point ξ and approaches u0 far from ξ as λ goes to zero. In particular, this shows that this problem admits more than one solution.

$$C^\infty $$ C ∞ Scaling Asymptotics for the Spectral Projector of the Laplacian

This article concerns new off-diagonal estimates on the remainder and its derivatives in the pointwise Weyl law on a compact n-dimensional Riemannian manifold. As an application, we prove that near any non-self-focal point, the scaling limit of the spectral projector of the Laplacian onto frequency windows of constant size is a normalized Bessel function depending only on n.

On Compact Manifolds with Harmonic Curvature and Positive Scalar Curvature

Let $$M^n(n\ge 3)$$ be an n-dimensional compact Riemannian manifold with harmonic curvature and positive scalar curvature. Assume that $$M^n$$ satisfies some integral pinching conditions. We give some rigidity theorems. In particular, Theorems 1.4 and 1.10 are sharp for our conditions have the additional properties of being sharp. By this, we mean that we can precisely characterize the case of equality.

Construction of Solutions for a Nonlinear Elliptic Problem on Riemannian Manifolds with Boundary

Abstract Let ( M , g ) ${(M,g)}$ be a smooth compact n-dimensional Riemannian manifold ( n ≥ 2 ${n\geq 2}$ ) with smooth ( n - 1 ) ${(n-1)}$ -dimensional boundary ∂ ⁡ M ${\partial M}$ . We prove that the stable critical points of the mean curvature of the boundary generates H 1 ⁢ ( M ) ${H^{1}(M)}$ solutions for the following singularly perturbed elliptic problem with Neumann boundary conditions: - ε 2 ⁢ Δ g ⁢ u + u = u p - 1 ⁢ in ⁢ M , u &gt; 0 ⁢ in ⁢ M , ∂ ⁡ u ∂ ⁡ ν = 0 ⁢ on ⁢ ∂ ⁡ M , $-\varepsilon^{2}\Delta_{g}u+u=u^{p-1}\text{ in }M,\quad u&gt;0\text{ in }M,\quad% \frac{\partial u}{\partial\nu}=0\text{ on }\partial M,$ when ε is small enough. Here p is subcritical.

Estimates of the first Steklov eigenvalue of properly embedded minimal hypersurfaces with free boundary

We consider a properly embedded minimal hypersurfacewith free boundary in a compact n-dimensional Riemannian manifold M be with nonnegative Ricci curvature and strictly convex boundary. Here, we obtain a new estimate from below for the first nonzero Steklov eigenvalue.

On a regularized family of models for the full Ericksen–Leslie system

We consider a general family of regularized systems for the full Ericksen–Leslie model for the hydrodynamics of liquid crystals in n-dimensional compact Riemannian manifolds. The system we consider consists of a regularized family of Navier–Stokes equations (including the Navier–Stokes-α-like equation, the Leray-α equation, the Modified Leray-α equation, the Simplified Bardina model, the Navier–Stokes–Voigt model and the Navier–Stokes equation) for the fluid velocity u suitably coupled with a parabolic equation for the director field d. We establish existence, stability and regularity results for this family. We also show the existence of a finite dimensional global attractor for our general model, and then establish sufficiently general conditions under which each trajectory converges to a single equilibrium by means of a Lojasiewicz–Simon inequality.

Regularized family of models for incompressible Cahn–Hilliard two-phase flows

We consider a general family of regularized models for incompressible two-phase flows based on the Cahn–Hilliard formulation in n-dimensional compact Riemannian manifolds (with or without boundary) for n=2,3. The system we consider consists of a regularized family of Navier–Stokes equations (including the Navier–Stokes-α-like model, the Leray-α model, the Modified Leray-α model, the Simplified Bardina model, the Navier–Stokes–Voight model, the Navier–Stokes model, and many others) for the fluid velocity u suitably coupled with a convective Cahn–Hilliard equation for the order (phase) parameter ϕ. We give a unified analysis of the entire three-parameter family of two-phase models. We first establish existence, stability and regularity results. Then, we show the existence of a global attractor and exponential attractor for our general model, and then establish precise conditions under which each trajectory (u,ϕ) converges to a single equilibrium by means of a Lojasiewicz–Simon inequality.

Bubbling solutions for supercritical problems on manifolds

Let (M,g) be an n-dimensional compact Riemannian manifold without boundary and Γ be a non-degenerate closed geodesic of (M,g). We prove that the supercritical problem−Δgu+hu=un+1n−3±ϵ,u>0, in (M,g) has a solution that concentrates along Γ as ϵ goes to zero, provided the function h and the sectional curvatures along Γ satisfy a suitable condition. A connection with the solution of a class of periodic Ordinary Differential Equations with singularity of attractive or repulsive type is established.

On the Ambrosetti–Malchiodi–Ni conjecture for general submanifolds

We study positive solutions of the following semilinear equationε2Δg¯u−V(z)u+up=0on M, where (M,g¯) is a compact smooth n-dimensional Riemannian manifold without boundary or the Euclidean space Rn, ε is a small positive parameter, p>1 and V is a uniformly positive smooth potential. Given k=1,…,n−1, and 1<p<n+2−kn−2−k. Assuming that K is a k-dimensional smooth, embedded compact submanifold of M, which is stationary and non-degenerate with respect to the functional ∫KVp+1p−1−n−k2dvol, we prove the existence of a sequence ε=εj→0 and positive solutions uε that concentrate along K. This result proves in particular the validity of a conjecture by Ambrosetti et al. [1], extending a recent result by Wang et al. [32], where the one co-dimensional case has been considered. Furthermore, our approach explores a connection between solutions of the nonlinear Schrödinger equation and f-minimal submanifolds in manifolds with density.