The classical Liouville theorem states that a bounded harmonic function on all of $\RR^n$ must be constant. In the early 1970s, S.T. Yau vastly generalized this, showing that it holds for manifolds with nonnegative Ricci curvature. Moreover, he conjectured a stronger Liouville property that has generated many significant developments. We will first discuss this conjecture and some of the ideas that went into its proof.
We will also discuss two recent areas where this circle of ideas has played a major role. One is Kleiner's new proof of Gromov's classification of groups of polynomial growth and the developments this generated. Another is to understanding singularities of mean curvature flow in high codimension. We will see that some of the ideas discussed in this survey naturally lead to a new approach to studying and classifying singularities of mean curvature flow in higher codimension. This is a subject that has been notoriously difficult and where much less is known than for hypersurfaces.
We study the wave analog of the Liouville equation and the constant mean curvature equations in 2 space dimensions, which are energy critical. We exhibit a blow-up criteria for the former using tools from conformal geometry, and we exhibit finite time blow-up for the latter under suitable assumptions on the initial data.
We prove some Liouville type theorems on smooth compact Riemannian manifolds with nonnegative sectional curvature and strictly convex boundary. This gives a nonlinear generalization in low dimension of the recent sharp lower bound of the first Steklov eigenvalue by Xia-Xiong and verifies partially a conjecture by the third author. As a consequence, we derive several sharp Sobolev trace inequalities on these manifolds.
In this note, we study Liouville type theorem for conformal Gaussian curvature equation (also called the mean field equation) $$ -Δu=K(x)e^u, in R^2 $$ where $K(x)$ is a smooth function on $R^2$. When $K(x)=K(x_1)$ is a sign-changing smooth function in the real line $R$, we have a non-existence result for the finite total curvature solutions. When $K$ is monotone non-decreasing along every ray starting at origin, we can prove a non-existence result too. We use moving plane method and moving sphere method.
In this paper we consider harmonic functions on gradient shrinking Ricci solitons with constant scalar curvature. A Liouville theorem is proved without using gradient estimate : any bounded harmonic function is constant on gradient shrinking Ricci solitons with constant scalar curvature. As an application, we show that the space of harmonic functions with polynomial growth has finite dimension.
Ancient solutions arise in the study of parabolic blow-ups. If we can categorize ancient solutions, we can better understand blow-up limits. Based on an argument of Giga and Kohn, we give a Liouville-type theorem restricting ancient, type-I, non-collapsing two- dimensional mean curvature flows to either spheres or cylinders.
We study some function-theoretic properties on a complete smooth metric measure space $(M,g,e^{-f}dv)$ with Bakry-Émery Ricci curvature bounded from below. We derive a Moser's parabolic Harnack inequality for the $f$-heat equation, which leads to upper and lower Gaussian bounds on the $f$-heat kernel. We also prove $L^p$-Liouville theorems in terms of the lower bound of Bakry-Émery Ricci curvature and the bound of function $f$, which generalize the classical Ricci curvature case and the $N$-Bakry-Émery Ricci curvature case.
We give a construction that connects the Cauchy problem for Liouville elliptic equation with a certain initial value problem for mean curvature one surfaces in hyperbolic 3-space H3, and solve both of them. We construct the only mean curvature one surface in H3 that passes through a given curve with given unit normal along it, and provide diverse applications. In particular, topics like period problems, symmetries, finite total curvature, planar geodesics, rigidity, etc. of surfaces are treated.
We analyze the probabilistic variance of a solution of Liouville's equation for curvature, given suitable bounds on the Gaussian curvature. The related systolic geometry was recently studied by Horowitz, Katz, and Katz, where we obtained a strengthening of Loewner's torus inequality containing a "defect term", similar to Bonnesen's strengthening of the isoperimetric inequality. Here the analogous isosystolic defect term depends on the metric and "measures" its deviation from being flat. Namely, the defect is the variance of the function f which appears as the conformal factor expressing the me…
In this paper, we will use the normalized intetral Ricci curvature to investigate Liouville type property of $ p $ harmonic function on Riemannian manifold. secondly, we will use the BiRic curvature to obtian Liuville theorem for $ p $ harmonic function or $ p $ harmonic 1 form. Lastly, we we will use the BiRic curvature to obtian Liouville theorem forharmonic $ q(q\geq 2)$ form.
Using minimal hypersurfaces, we obtain topological obstructions to admitting complete metrics with positive scalar curvature on a given class of non-compact n-manifolds with n less than 8. We show that the Liouville theorem for a locally conformally flat n-manifold of non-negative scalar curvature follows from the impossibility of there being a positive scalar curvature metric on its connect sum with the n-torus. With the recent work of Chodosh-Li, the Liouville theorem is now proved in all remaining cases. Finally, using MOTS instead of minimal hypersurfaces, we show an Initial Data Set versi…
We prove that for combinatorial graphs with non-negative Ollivier curvature, one has \[ \|P_t μ- P_t ν\|_1 \leq \frac{W_1(μ,ν)}{\sqrt{t}} \] for all probability measures $μ,ν$ where $P_t$ is the heat semigroup and $W_1$ is the $\ell_1$-Wasserstein distance. This turns out to be an equivalent formulation of a version of reverse Poincaré inequality. Furthermore, this estimate allows us to prove Buser inequality, Liouville property and the the eigenvalue estimate $λ_1 \geq \log(2)/\operatorname{diam}^2$.
In this paper we classify the solutions to the geometric Neumann problem for the Liouville equation in the upper half-plane or an upper half-disk, with the energy condition given by finite area. As a result, we classify the conformal Riemannian metrics of constant curvature and finite area on a half-plane that have a finite number of boundary singularities, not assumed a priori to be conical, and constant geodesic curvature along each boundary arc.
In this article, we prove a Liouville property of holomorphic maps from a complete Kahler manifold with nonnegative holomorphic bisectional curvature to a complete simply connected Kahler manifold with a certain assumption on the sectional curvature.
In this paper, we generalize Liouville type theorems for some semilinear partial differential inequalities to sub-Riemannian manifolds satisfying a nonnegative generalized curvature-dimension inequality introduced by Baudoin and Garofalo in [5]. In particular, our results apply to all Sasakian manifolds with nonnegative horizontal Webster-Tanaka-Ricci curvature. The key ingredient is to construct a class of "good" cut-off functions. We also provide some upper bounds for lifespan to parabolic and hyperbolic inequalities.
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 \(2<p<n\).
We prove a Liouville theorem for the plurisubharmonic functions on complete Kaelher manifolds. As the applications, we prove a splitting theorem for complete Kaehler manifolds with nonnegative biscetional curvature in terms of the linear growth harmonic functions and a optomal gap theorem for such manifolds.
Two necessary conditions for the induced metrics of parallel mean curvature surfaces in a complex space form of complex two-dimension are observed. One is similar to the Ricci condition of the classical surface theory in Euclidean three-space and the other is related to the Liouville metric. Conversely, we prove that a special type of the Liouville metric on a domain in the Euclidean two-plane whose Gaussian curvature satisfies the differential equation similar to the Ricci condition is explicitly determined by an elliptic function. We have isometric immersions from a simply connected two-dime…
We derive some consequences of the Liouville theorem for plurisubharmonic functions of L.-F. Tam and the author. The first result provides a nonlinear version of the complex splitting theorem (which splits off a factor of $\mathbb{C}$ isometrically from the simply-connected Kähler manifold with nonnegative bisectional curvature and a linear growth holomorphic function) of L.-F. Tam and the author. The second set of results concerns the so-called $k$-hyperbolicity and its connection with the negativity of the $k$-scalar curvature (when $k=1$ they are the negativity of holomorphic sectional curv…
We investigate different concentration-compactness phenomena related to the Q-curvature in arbitrary even dimension. We first treat the case of an open domain in $R^{2m}$, then that of a closed manifold and, finally, the particular case of the sphere $S^{2m}$. In all cases we allow the sign of the Q-curvature to vary, and show that in the case of a closed manifold, contrary to the case of open domains in $R^{2m}$, concentration phenomena can occur only at points of positive Q-curvature. As a consequence, on a locally conformally flat manifold of non-positive Euler characteristic we always have…
In this paper, we first obtain an $L^q$ gradient estimate for $p$-harmonic maps, by assuming the target manifold supporting a certain function, whose gradient and Hessian satisfy some analysis conditions. From this $L^q$ gradient estimate, we get a corresponding Liouville type result for $p$-harmonic maps. Secondly, using these general results, we give various geometric applications to $p$-harmonic maps from complete manifolds with nonnegative Ricci curvature to manifolds with various upper bound on sectional curvature, under appropriate controlled images.
We prove uniqueness of solutions for the nonlocal Liouville equation $$ (-Δ)^{1/2} w = K e^w \quad \mbox{in $\mathbb{R}$} $$ with finite total $Q$-curvature $\int_{\mathbb{R}} K e^w \, dx< +\infty$. Here the prescribed $Q$-curvature function $K=K(|x|) > 0$ is assumed to be a positive, symmetric-decreasing function satisfying suitable regularity and decay bounds. In particular, we obtain uniqueness of solutions in the Gaussian case with $K(x) = \exp(-x^2)$. Our uniqueness proof exploits a connection of the nonlocal Liouville equation to ground state solitons for Calogero--Moser derivative NLS, …
We classify the solutions to the equation (- Δ)^m u=(2m-1)!e^{2mu} on R^{2m} giving rise to a metric g=e^{2u}g_{R^{2m}} with finite total $Q$-curvature in terms of analytic and geometric properties. The analytic conditions involve the growth rate of u and the asymptotic behaviour of Δu(x) as |x|\to \infty. As a consequence we give a geometric characterization in terms of the scalar curvature of the metric e^{2u}g_{R^{2m}} at infinity, and we observe that the pull-back of this metric to $S^{2m}$ via the stereographic projection can be extended to a smooth Riemannian metric if and only if it is …
In this paper, we study the convexity, interior gradient estimate, Liouville type theorem and asymptotic behavior at infinity of translating solutions to mean curvature flow as well as the nonlinear flow by powers of the mean curvature.
We study metrics of constant $Q$-curvature in the Euclidean space with a prescribed singularity at the origin, namely solutions to the equation $$(-Δ)^\frac{n}{2}w=e^{nw}-cδ_{0} \text{ on } \mathbb R^n,$$ under a finite volume condition. We analyze the asymptotic behaviour at infinity and the existence of solutions for every $n\ge 3$ also in a supercritical regime. Finally, we state some open problems.
In this paper, we will study harmonic functions on the complete and incomplete spaces with nonnegative Ricci curvature which exhibit inhomogeneous collapsing behaviors at infinity. The main result states that any nonconstant harmonic function on such spaces yields a definite exponential growth rate which depends explicitly on the geometric data at infinity.
We estimate from below by geometric data the eigenvalues of the periodic Sturm-Liouville operator $- 4 d^2/ds^2 + κ^2 (s)$ with potential given by the curvature of a closed curve.
The Ergodicity of Geodesic Flows on Rank One Manifolds of Nonpositive Curvature
College of Mathematics and Systems Science, Shandong University of Science and Technology（山东科技大学数学与系统科学学院）；Department of Mathematics, Southern University of Science and Technology（南方科技大学数学系）
We define a formal Riemannian metric on a given conformal class of metrics on a closed Riemann surface. We show interesting formal properties for this metric, in particular the curvature is nonpositive and the Liouville energy is geodesically convex. The geodesic equation for this metric corresponds to a degenerate elliptic fully nonlinear PDE, and we prove that any two points are connected by a $C^{1,1}$ geodesic. Using this we can define a length space structure on the given conformal class. We present a different approach to the uniformization theorem by studying the negative gradient flow …
In this paper, we will address to the following parabolic equation $$ u_t=Δ_fu + F(u) $$ on a smooth metric measure space with Bakry-Émery curvature bounded from below. Here $F$ is a differentiable function defined in $\mathbb{R}$. Our motivation is originally inspired by gradient estimates of Allen-Cahn and Fisher equations (\cite{Bai17, CLPW17}). In this paper, we show new gradient estimates for these equations. As their applications, we obtain Liouville type theorems for positive or bounded solutions to the above equation when either $F=cu(1-u)$ (the Fisher equation) or; $F=-u^3+u$ (the All…
We introduce and study an approximate solution of the p-Laplace equation, and a linearlization $L_ε$ of a perturbed p-Laplace operator. By deriving an $L_ε$-type Bochner's formula and a Kato type inequality, we prove a Liouville type theorem for weakly p-harmonic functions with finite p-energy on a complete noncompact manifold M which supports a weighted Poincaré inequality and satisfies a curvature assumption. This nonexistence result, when combined with an existence theorem, yields in turn some information on topology, i.e. such an M has at most one p-hyperbolic end. Moreover, we prove a Lio…
