Prescribed scalar curvature plus mean curvature flows in compact manifolds with boundary of negative conformal invariant

Using a geometric flow, we study the following prescribed scalar curvature plus mean curvature problem: Let \((M,g_0)\) be a smooth compact manifold of dimension \(n\ge 3\) with boundary. Given any smooth functions f in M and h on \(\partial M\), does there exist a conformal metric of \(g_0\) such that its scalar curvature equals f and boundary mean curvature equals h? Assume that f and h are negative and the conformal invariant \(Q(M,\partial M)\) is a negative real number, we prove the global existence and convergence of the so-called prescribed scalar curvature plus mean curvature flows. Via a family of such flows together with some additional variational arguments, we prove the existence and uniqueness of positive minimizers of the associated energy functional and give a confirmative answer to the above problem. The same result also can be obtained by sub–super-solution method and subcritical approximations.     

Geometric rigidity of constant heat flow

Let \(\Omega \) be a compact Riemannian manifold with smooth boundary and let \(u_t\) be the solution of the heat equation on \(\Omega \), having constant unit initial data \(u_0=1\) and Dirichlet boundary conditions (\(u_t=0\) on the boundary, at all times). If at every time t the normal derivative of \(u_t\) is a constant function on the boundary, we say that \(\Omega \) has the constant flow property. This gives rise to an overdetermined parabolic problem, and our aim is to classify the manifolds having this property. In fact, if the metric is analytic, we prove that \(\Omega \) has the constant flow property if and only if it is an isoparametric tube, that is, it is a solid tube of constant radius around a closed, smooth, minimal submanifold, with the additional property that all equidistants to the boundary (parallel hypersurfaces) are smooth and have constant mean curvature. Hence, the constant flow property can be viewed as an analytic counterpart to the isoparametric property. Finally, we relate the constant flow property with other overdetermined problems, in particular, the well-known Serrin problem on the mean-exit time function, and discuss a counterexample involving minimal free boundary immersions into Euclidean balls.     

A Liouville-type theorem for biharmonic maps between complete Riemannian manifolds with small energies

We prove a Liouville-type theorem for biharmonic maps from a complete Riemannian manifold of dimension \(n\) that has a lower bound on its Ricci curvature and positive injectivity radius into a Riemannian manifold whose sectional curvature is bounded from above. Under these geometric assumptions we show that if the \(L^p\)-norm of the tension field is bounded and the n-energy of the map is sufficiently small, then every biharmonic map must be harmonic, where \(2<p<n\).     

Large conformal metrics with prescribed sign-changing Gauss curvature

Let \((M,g)\) be a two dimensional compact Riemannian manifold of genus \(g(M)>1\). Let \(f\) be a smooth function on \(M\) such that

$$\begin{aligned} f \ge 0, \quad f\not \equiv 0, \quad \min _M f = 0. \end{aligned}$$

Let \(p_1,\ldots ,p_n\) be any set of points at which \(f(p_i)=0\) and \(D^2f(p_i)\) is non-singular. We prove that for all sufficiently small \(\lambda >0\) there exists a family of “bubbling” conformal metrics \(g_\lambda =e^{u_\lambda }g\) such that their Gauss curvature is given by the sign-changing function \(K_{g_\lambda }=-f+\lambda ^2\). Moreover, the family \(u_\lambda \) satisfies

$$\begin{aligned} u_\lambda (p_j) = -4\log \lambda -2\log \left( \frac{1}{\sqrt{2}} \log \frac{1}{\lambda }\right) +O(1) \end{aligned}$$

and

$$\begin{aligned} \lambda ^2e^{u_\lambda }\rightharpoonup 8\pi \sum _{i=1}^{n}\delta _{p_i},\quad \text{ as } \lambda \rightarrow 0, \end{aligned}$$

where \(\delta _{p}\) designates Dirac mass at the point \(p\).

     

Existence of self-Cheeger sets on Riemannian manifolds

Let \(({\mathcal M},g)\) be a smooth compact Riemannian manifold of dimension \(N\ge 2\). We prove the existence of a family \((\Omega _\varepsilon )_{\varepsilon \in (0,\varepsilon _0)}\) of self-Cheeger sets in \(({\mathcal M},g)\). The domains \(\Omega _\varepsilon \subset {\mathcal M}\) are perturbations of geodesic balls of radius \(\varepsilon \) centered at \(p \in {\mathcal M}\), and in particular, if \(p_0\) is a non-degenerate critical point of the scalar curvature of g, then the family \((\partial \Omega _\varepsilon )_{\varepsilon \in (0,\varepsilon _0)}\) constitutes a smooth foliation of a neighborhood of \(p_0\).     

Blowup behavior of harmonic maps with finite index

In this paper, we study the blow-up phenomena on the \(\alpha _k\)-harmonic map sequences with bounded uniformly \(\alpha _k\)-energy, denoted by \(\{u_{\alpha _k}: \alpha _k>1 \quad \text{ and } \quad \alpha _k\searrow 1\}\), from a compact Riemann surface into a compact Riemannian manifold. If the Ricci curvature of the target manifold has a positive lower bound and the indices of the \(\alpha _k\)-harmonic map sequence with respect to the corresponding \(\alpha _k\)-energy are bounded, then we can conclude that, if the blow-up phenomena occurs in the convergence of \(\{u_{\alpha _k}\}\) as \(\alpha _k\searrow 1\), the limiting necks of the convergence of the sequence consist of finite length geodesics, hence the energy identity holds true. For a harmonic map sequence \(u_k:(\Sigma ,h_k)\rightarrow N\), where the conformal class defined by \(h_k\) diverges, we also prove some similar results.     

Heinz mean curvature estimates in warped product spaces $$M\times _{e^{\psi }}N$$

If a graph submanifold (x, f(x)) of a Riemannian warped product space \((M^m\times _{e^{\psi }}N^n,\tilde{g}=g+ e^{2\psi }h)\) is immersed with parallel mean curvature H, then we obtain a Heinz-type estimation of the mean curvature. Namely, on each compact domain D of M, \(m\Vert H\Vert \le \frac{A_{\psi }(\partial D)}{V_{\psi }(D)}\) holds, where \(A_{\psi }(\partial D)\) and \(V_{\psi }(D)\) are the \({\psi }\)-weighted area and volume, respectively. In particular, \(H=0\) if (M, g) has zero-weighted Cheeger constant, a concept recently introduced by Impera et al. (Height estimates for killing graphs. arXiv:1612.01257, 2016). This generalizes the known cases \(n=1\) or \(\psi =0\). We also conclude minimality using a closed calibration, assuming \((M,g_*)\) is complete where \(g_*=g+e^{2\psi }f^*h\), and for some constants \(\alpha \ge \delta \ge 0\), \(C_1>0\) and \(\beta \in [0,1)\), \(\Vert \nabla ^*\psi \Vert ^2_{g_*}\le \delta \), \(\mathrm {Ricci}_{\psi ,g_*}\ge \alpha \), and \({\mathrm{det}}_g(g_*)\le C_1 r^{2\beta }\) holds when \(r\rightarrow +\infty \), where r(x) is the distance function on \((M,g_*)\) from some fixed point. Both results rely on expressing the squared norm of the mean curvature as a weighted divergence of a suitable vector field.     

Eigenvalue Estimate for the Basic Laplacian on Manifolds with Foliated Boundary,Part II

On a compact Riemannian manifold N whose boundary is endowed with a Riemannian flow, we gave in El Chami et al. (Eigenvalue estimate for the basic Laplacian on manifolds with foliated boundary, 2015) a sharp lower bound for the first non-zero eigenvalue of the basic Laplacian acting on basic 1-forms. In this paper, we extend this result to the set of basic p-forms when \(p>1\). We then characterize the limiting case by showing that the manifold N is isometric to Open image in new window for some group \(\Gamma \) where \(B'\) denotes the unit closed ball. As a consequence, we describe the Riemannian product \({\mathbb {S}}^1\times {\mathbb {S}}^n\) as the boundary of a manifold.     

The Dirichlet problem for Einstein metrics on cohomogeneity one manifolds

Let \(G{/}H\) be a compact homogeneous space, and let \(\hat{g}_0\) and \(\hat{g}_1\) be G-invariant Riemannian metrics on \(G/H\). We consider the problem of finding a G-invariant Einstein metric g on the manifold \(G/H\times [0,1]\) subject to the constraint that g restricted to \(G{/}H\times \{0\}\) and \(G/H\times \{1\}\) coincides with \(\hat{g}_0\) and \(\hat{g}_1\), respectively. By assuming that the isotropy representation of \(G/H\) consists of pairwise inequivalent irreducible summands, we show that we can always find such an Einstein metric.     

On the approximation of dynamical indicators in systems with nonuniformly hyperbolic behavior

Let f be a \(C^{1+\alpha }\) diffeomorphism of a compact Riemannian manifold and \(\mu \) an ergodic hyperbolic measure with positive entropy. We prove that for every continuous potential \(\phi \) there exists a sequence of basic sets \(\Omega _n\) such that the topological pressure \(P(f|\Omega _n,\phi )\) converges to the free energy \(P_{\mu }(\phi ) = h(\mu ) + \int \phi {d\mu }\). We also prove that for a suitable class of potentials \(\phi \) there exists a sequence of basic sets \(\Omega _n\) such that \(P(f|\Omega _n,\phi ) \rightarrow P(\phi )\).     

Multiplicity of nodal solutions to the Yamabe problem

Given a compact Riemannian manifold (M, g) without boundary of dimension \(m\ge 3\) and under some symmetry assumptions, we establish existence of one positive and multiple nodal solutions to the Yamabe-type equation

$$\begin{aligned} -\text {div}_{g}(a\nabla u)+bu=c|u|^{2^{*}-2}u\quad \text { on }M, \end{aligned}$$

where \(a,b,c\in \mathcal {C}^{\infty }(M), a\) and c are positive, ? div\(_{g}(a\nabla )+b\) is coercive, and \(2^{*}=\frac{2m}{m-2}\) is the critical Sobolev exponent. In particular, if \(R_{g}\) denotes the scalar curvature of (M, g), we give conditions which guarantee that the Yamabe problem

$$\begin{aligned} \Delta _{g}u+\frac{m-2}{4(m-1)}R_{g}u=\kappa u^{2^{*}-2}\quad \text { on }M \end{aligned}$$

admits a prescribed number of nodal solutions.

     

CMC Foliations of Closed Manifolds

We prove that every closed, smooth \(n\)-manifold \(X\) admits a Riemannian metric together with a constant mean curvature (CMC) foliation if and only if its Euler characteristic is zero, where by a CMC foliation we mean a smooth, codimension-one, transversely oriented foliation with leaves of CMC and where the value of the CMC can vary from leaf to leaf. Furthermore, we prove that this CMC foliation of \(X\) can be chosen so that when \(n\ge 2\), the constant values of the mean curvatures of its leaves change sign. We also prove a general structure theorem for any such non-minimal CMC foliation of \(X\) that describes relationships between the geometry and topology of the leaves, including the property that there exist compact leaves for every attained value of the mean curvature.     

A stochastic Gauss–Bonnet–Chern formula

We prove that a Gaussian ensemble of smooth random sections of a real vector bundle \(E\) over compact manifold \(M\) canonically defines a metric on \(E\) together with a connection compatible with it. Additionally, we prove a refined Gauss-Bonnet theorem stating that if the bundle \(E\) and the manifold \(M\) are oriented, then the Euler form of the above connection can be identified, as a current, with the expectation of the random current defined by the zero-locus of a random section in the above Gaussian ensemble.     

Some identities for Appell–Lerch sums and a universal mock theta function

In their paper “A survey of classical mock theta functions”, Gordon and McIntosh observed that the classical mock \(\theta \)-functions, including those found by Ramanujan, can be expressed in terms of two ‘universal’ mock \(\theta \)-functions denoted by \(g_{_{2}}\) and \(g_{_{3}}\). These functions are normalized level 2 and level 3 Appell–Lerch functions. In the survey paper, the authors list several identities for certain Appell–Lerch functions and refer the proofs to a future paper with this title, listed in their references as [GM3]. The purpose of this paper is to prove these identities. One of the identities removes the \( \theta \) -quotients from Kang’s formulas, which express \(g_{_{2}}\) and \({g}_{{_{3}}}\) in terms of Zwegers’ \(\mu \)-function and \( \theta \)-quotients.     

Estimates for the first eigenvalue of Jacobi operator on hypersurfaces with constant mean curvature in spheres

In this paper, we study the first eigenvalue of Jacobi operator on an n-dimensional non-totally umbilical compact hypersurface with constant mean curvature H in the unit sphere \(S^{n+1}(1)\). We give an optimal upper bound for the first eigenvalue of Jacobi operator, which only depends on the mean curvature H and the dimension n. This bound is attained if and only if, \(\varphi :\ M \rightarrow S^{n+1}(1)\) is isometric to \(S^1(r)\times S^{n-1}(\sqrt{1-r^2})\) when \(H\ne 0\) or \(\varphi :\ M \rightarrow S^{n+1}(1)\) is isometric to a Clifford torus \( S^{n-k}\left( \sqrt{\dfrac{n-k}{n}}\right) \times S^k\left( \sqrt{\dfrac{k}{n}}\right) \), for \(k=1, 2, \ldots , n-1\) when \(H=0\).     

Reflecting Diffusion Processes on Manifolds Carrying Geometric Flow

Let \(L_t:=\Delta _t+Z_t\) for a \(C^{\infty }\)-vector field Z on a differentiable manifold M with boundary \(\partial M\), where \(\Delta _t\) is the Laplacian operator, induced by a time dependent metric \(g_t\) differentiable in \(t\in [0,T_\mathrm {c})\). We first establish the derivative formula for the associated reflecting diffusion semigroup generated by \(L_t\). Then, by using parallel displacement and reflection, we construct the couplings for the reflecting \(L_t\)-diffusion processes, which are applied to gradient estimates and Harnack inequalities of the associated heat semigroup. Finally, as applications of the derivative formula, we present a number of equivalent inequalities for a new curvature lower bound and the convexity of the boundary. These inequalities include the gradient estimates, Harnack inequalities, transportation-cost inequalities and other functional inequalities for diffusion semigroups.     

Rigidity of Free Boundary Surfaces in Compact 3-Manifolds with Strictly Convex Boundary

In this paper, we obtain an analogue of Toponogov theorem in dimension 3 for compact manifolds \(M^3\) with nonnegative Ricci curvature and strictly convex boundary \(\partial M\). Here we obtain a sharp upper bound for the length \(L(\partial \Sigma )\) of the boundary \(\partial \Sigma \) of a free boundary minimal surface \(\Sigma ^2\) in \(M^3\) in terms of the genus of \(\Sigma \) and the number of connected components of \(\partial \Sigma \), assuming \(\Sigma \) has index one. After, under a natural hypothesis on the geometry of M along \(\partial M\), we prove that if \(L(\partial \Sigma )\) saturates the respective upper bound, then \(M^3\) is isometric to the Euclidean 3-ball and \(\Sigma ^2\) is isometric to the Euclidean disk. In particular, we get a sharp upper bound for the area of \(\Sigma \), when \(M^3\) is a strictly convex body in \(\mathbb {R}^3\), which is saturated only on the Euclidean 3-balls (by the Euclidean disks). We also consider similar results for free boundary stable CMC surfaces.     

Bubbling analysis for approximate Lorentzian harmonic maps from Riemann surfaces

For a sequence of approximate harmonic maps \((u_n,v_n)\) (meaning that they satisfy the harmonic system up to controlled error terms) from a compact Riemann surface with smooth boundary to a standard static Lorentzian manifold with bounded energy, we prove that identities for the Lorentzian energy hold during the blow-up process. In particular, in the special case where the Lorentzian target metric is of the form \(g_N -\beta dt^2\) for some Riemannian metric \(g_N\) and some positive function \(\beta \) on N, we prove that such identities also hold for the positive energy (obtained by changing the sign of the negative part of the Lorentzian energy) and there is no neck between the limit map and the bubbles. As an application, we complete the blow-up picture of singularities for a harmonic map flow into a standard static Lorentzian manifold. We prove that the energy identities of the flow hold at both finite and infinite singular times. Moreover, the no neck property of the flow at infinite singular time is true.     

Rigidity of complete manifolds with parallel Cotton tensor

The aim of this paper is to show some rigidity results for complete Riemannian manifolds with parallel Cotton tensor. In particular, we prove that any compact manifold of dimension \(n\ge 3\) with parallel Cotton tensor and positive constant scalar curvature is isometric to a finite quotient of \({\mathbb {S}}^n\) under a pointwise or integral pinching condition. Moreover, a rigidity theorem for stochastically complete manifolds with parallel Cotton tensor is also given. The proofs rely mainly on curvature elliptic estimates and the weak maximum principle.