We classify noncompact homogeneous spaces which are Einstein and asymptotically harmonic. This completes the classification of Riemannian harmonic spaces in the homogeneous case: Any simply connected homogeneous harmonic space is flat, or rank-one symmetric, or a nonsymmetric Damek-Ricci space. Independently, Y. Nikolayevsky has obtained the latter classification under the additional assumption of nonpositive sectional curvatures.
In the present paper, we investigate Busemann functions in a general Finsler setting as well as in asymptotically harmonic Finsler manifolds. In particular, we show Busemann functions are smooth on asymptotically harmonic Finsler manifolds.
Let (M,g) be a complete, simply connected Riemannian manifold of dimension 3 without conjugate points. We show that M is a hyperbolic manifold of constant sectional curvature, provided M is asymptotically harmonic of constant h > 0.
Generalizing the result of Li and Tam for the hyperbolic spaces, we prove an existence theorem on the Dirichlet problem for harmonic maps with $C^1$ boundary conditions at infinity between asymptotically hyperbolic manifolds.
Let $(M,g)$ be a complete, simply connected Riemannian manifold of dimension 3 without conjugate points. We show that $M$ is a flat manifold, provided $M$ is asymptotically harmonic of constant $h = 0$.
In this article we consider asymptotically harmonic manifolds which are simply connected complete Riemannian manifolds without conjugate points such that all horospheres have the same constant mean curvature $h$. We prove the following equivalences for asymptotically harmonic manifolds $X$ under the additional assumption that their curvature tensor together with its covariant derivative are uniformly bounded: (a) $X$ has rank one; (b) $X$ has Anosov geodesic flow; (c) $X$ is Gromov hyperbolic; (d) $X$ has purely exponential volume growth with volume entropy equals $h$. This generalizes earlier…
Asymptotically harmonic manifolds are simply connected complete Riemannian manifolds without conjugate points such that all horospheres have the same constant mean curvature $h$. In this article we present results for harmonic functions on rank one asymptotically harmonic manifolds $X$ with mild curvature boundedness conditions. Our main results are (a) the explicit calculation of the Radon-Nykodym derivative of the visibility measures, (b) an explicit integral representation for the solution of the Dirichlet problem at infinity in terms of these visibility measures, and (c) a result on horosp…
We study asymptotically harmonic manifolds of negative curvature, without any cocompactness or homogeneity assumption. We show that asymptotic harmonicity provides a lot of information on the asymptotic geometry of these spaces: in particular, we determine the volume entropy, the spectrum and the relative densities of visual and harmonic measures on the ideal boundary. Then, we prove an asymptotic analogue of the classical mean value property of harmonic manifolds, and we characterize asymptotically harmonic manifolds, among Cartan-Hadamard spaces of strictly negative curvature, by the existen…
We use Busemann functions to construct volume preserving mappings in an asymptotically harmonic manifold. If the asymptotically harmonic manifold satisfies the visibility condition, we construct mappings which preserve distances in some directions. We also prove that some integrals on the intersection of horospheres are independent of the differences between the values of the corresponding Busemann functions and we establish an upper bound of the volume of the intersection of two horospheres which is independent of the difference between values of corresponding Busemann functions.
Our purpose is to use a Darboux homogenous derivative to understand the harmonic maps with values in homogeneous space. We present a characterization of these harmonic maps from the geometry of homogeneous space. Furthermore, our work covers all type of invariant geometry in homogeneous space.
We prove local existence of complex-valued harmonic morphisms from any Riemannian homogeneous spaces of positive curvature, except the Berger space Sp(2)/SU(2).
We present a new method for manufacturing complex-valued harmonic morphisms from a wide class of Riemannian Lie groups. This yields new solutions from an important family of homogeneous Hadamard manifolds. We also give a new method for constructing left-invariant foliations on a large class of Lie groups producing harmonic morphisms
In this work we study properties of stability and non-stability of harmonic maps under the homogeneous Ricci flow. We provide examples where the stability (non-stability) is preserved under the Ricci flow and an example where the Ricci flow does not preserve the stability of an harmonic map.
A Riemannian manifold is called harmonic if its volume density function expressed in polar coordinates centered at any point is radial. Flat and rank-one symmetric spaces are harmonic. The converse (the Lichnerowicz Conjecture) is true for manifolds of nonnegative scalar curvature and for some other classes of manifolds, but is not true in general: there exists a family of homogeneous harmonic spaces, the Damek-Ricci spaces, containing noncompact rank-one symmetric spaces, as well as infinitely many nonsymmetric examples. We prove that a harmonic homogeneous manifold of nonpositive curvature i…
Consider an asymptotically flat Riemannian manifold $(M,g)$ of dimension $n \geq 3$ with nonempty compact boundary. We recall the harmonic conformal class $[g]_h$ of the metric, which consists of all conformal rescalings given by a harmonic function raised to an appropriate power. The geometric significance is that every metric in $[g]_h$ has the same pointwise sign of scalar curvature. For this reason, the harmonic conformal class appears in the study of general relativity, where scalar curvature is related to energy density. Our purpose is to introduce and study invariants of the harmonic co…
Asymptotic behavior of energy of a harmonic map defined on an asymptotically hyperbolic manifold is considered. Using the growth of energy, we show that a harmonic map defined on some asymptotically hyperbolic manifolds has to be constant if the total energy is finite, or if the map approaches a point fast enough, in terms of a defining function for the boundary.
A complete Riemannian manifold without conjugate points is called asymptotically harmonic if the mean curvature of its horospheres is a universal constant. Examples of asymptotically harmonic manifolds include flat spaces and rank one locally symmetric spaces of noncompact type. In this paper we show that this list exhausts the compact asymptotically harmonic manifolds under a variety of assumptions including nonpositive curvature or Gromov hyperbolic fundamental group. We then present a new characterization of symmetric spaces amongst the set of all visibility manifolds
We consider weakly and strongly asymptotically mean value harmonic (amv-harmonic) functions on subriemannian and RCD settings. We demonstrate that, in non-collapsed RCD-spaces with vanishing metric measure boundary, Cheeger harmonic functions are weakly amv-harmonic and that, in Carnot groups, weak amv-harmonicity equivalently characterizes harmonicity in the sense of the sub-Laplacian. In homogeneous Carnot groups of step $2$, we prove a Blaschke-Privaloff-Zaremba type theorem. Similar results are discussed in the settings of Riemannian manifolds and for Alexandrov surfaces.
An explicit lower bound for the mass of an asymptotically flat Riemannian 3-manifold is given in terms of linear growth harmonic functions and scalar curvature. As a consequence, a new proof of the positive mass theorem is achieved in dimension three. The proof has parallels with both the Schoen-Yau minimal hypersurface technique and Witten's spinorial approach. In particular, the role of harmonic spinors and the Lichnerowicz formula in Witten's argument is replaced by that of harmonic functions and a formula introduced by the fourth named author in recent work, while the level sets of harmoni…
$(M^n,g)$ be a complete Riemannian manifold without conjugate points. In this paper, we show that if $M$ is also simply connected, then $M$ is flat, provided that $M$ is also asymptotically harmonic manifold with minimal horospheres (AHM). The (first order) flatness of $M$ is shown by using the strongest criterion: $\{{e_i}\}$ be an orthonormal basis of $T_{p}M$ and $\{b_{e_{i}}\}$ be the corresponding Busemann functions on $M$. Then, (1) The vector space $V = span\{b_{v} | v \in T_{p}M \}$ is finite dimensional and dim $V = $ dim $M = n$.(2) $\{\nabla b_{e_i}(p) \}$ is a global parallel ortho…
In this paper we focus on the uniqueness question for (expanding) solutions of the Harmonic map flow coming out of smooth 0-homogeneous maps with values into a closed Riemannian manifold. We introduce a relative entropy for two purposes. On the one hand, we prove the existence of two expanding solutions associated to any suitable solution coming out of a $0$-homogeneous map by a blow-up and a blow-down process. On the other hand, generic uniqueness of expanding solutions coming out of the same 0-homogeneous map of 0 relative entropy is proved.
A Riemannian manifold is called geometrically formal if the wedge product of harmonic forms is again harmonic, which implies in the compact case that the manifold is topologically formal in the sense of rational homotopy theory. A manifold admitting a Riemannian metric of positive sectional curvature is conjectured to be topologically formal. Nonetheless, we show that among the homogeneous Riemannian metrics of positive sectional curvature a geometrically formal metric is either symmetric, or a metric on a rational homology sphere.
In this paper we construct a new class of harmonic and asymptotically harmonic Finsler manifolds of $(α,β)$-type. This class is defined by a Riemannian metric $α$ and a special 1-form $β$.
Let G be a compact, semi-simple Lie group and H a maximal rank reductive subgroup. The irreducible representations of G can be constructed as spaces of harmonic spinors with respect to a Dirac operator on the homogeneous space G/H twisted by bundles associated to the irreducible, possibly projective, representations of H. Here, we give a quick proof of this result, computing the index and kernel of this twisted Dirac operator using a homogeneous version of the Weyl character formula noted by Gross, Kostant, Ramond, and Sternberg, as well as recent work of Kostant regarding an algebraic version…
It is well-known that for a surface in a 3-dimensional real space form the constancy of the mean curvature is equivalent to the harmonicity of the Gauss map. However, this is not true in general for surfaces in an arbitrary 3-dimensional ambient space. In this paper we study this problem for surfaces in an important and very natural family of 3-dimensional ambient spaces, namely homogeneous spaces. In particular, we obtain a full classification of constant mean curvature surfaces, whose Gauss map satisfies the more mild condition of vertical harmonicity, in all 3-dimensional homogeneous spaces…
In this paper, we study stability for harmonic foliations on locally conformal Kähler manifolds with complex leaves. We also discuss instability for harmonic foliations on compact submanifolds immersed in Euclidean spaces and compact homogeneous spaces.
We study regularity properties of solutions to the Dirichlet problem for the complex Homogeneous Monge-Ampère equation. We show that for certain boundary data on $\mathbb P^1$ the solution $Φ$ to this Dirichlet problem is connected via a Legendre transform to an associated flow in the complex plane called the Hele-Shaw flow. Using this we determine precisely the harmonic discs associated to $Φ$. We then give examples for which these discs are not dense in the product, and also prove that this situation persists after small perturbations of the boundary data.
We study the topology of a complete asymptotically hyperbolic Einstein manifold such that its conformal boundary has positive Yamabe invariant. We proved that all maps from such manifold into any nonpositively curved manifold are homotopically trivial. Our proof is based on a Bochner type argument on harmonic maps.
An infinite family of distinct harmonic morphisms with minimal circle fibers from the 7-dimensional homogeneous Allof-Wallach spaces of positive curvature onto the 6-dimensional flag manifolds is given.
We consider Legendrian contact structures on odd-dimensional complex analytic manifolds. We are particularly interested in integrable structures, which can be encoded by compatible complete systems of second order PDEs on a scalar function of many independent variables and considered up to point transformations. Using the techniques of parabolic differential geometry, we compute the associated regular, normal Cartan connection and give explicit formulas for the harmonic part of the curvature. The PDE system is trivializable by means of point transformations if and only if the harmonic curvatur…
We give a lower bound for the Lorentz length of the ADM energy-momentum vector (ADM mass) of 3-dimensional asymptotically flat initial data sets for the Einstein equations. The bound is given in terms of linear growth `spacetime harmonic functions' in addition to the energy-momentum density of matter fields, and is valid regardless of whether the dominant energy condition holds or whether the data possess a boundary. A corollary of this result is a new proof of the spacetime positive mass theorem for complete initial data or those with weakly trapped surface boundary, and includes the rigidity…
The third real de Rham cohomology of compact homogeneous spaces is studied. Given $M=G/K$ with $G$ compact semisimple, we first show that each bi-invariant symmetric bilinear form $Q$ on $\mathfrak{g}$ such that $Q|_{\mathfrak{k}\times\mathfrak{k}}=0$ naturally defines a $G$-invariant closed $3$-form $H_Q$ on $M$, which plays the role of the so called Cartan $3$-form $Q([\cdot,\cdot],\cdot)$ on the compact Lie group $G$. Indeed, every class in $H^3(G/K)$ has a unique representative $H_Q$. Secondly, focusing on the class of homogeneous spaces with the richest third cohomology (other than Lie gr…
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.
In the pioneering work of Stern, level sets of harmonic functions have been shown to be an effective tool in the study of scalar curvature in dimension 3. Generalizations of this idea, utilizing level sets of so called spacetime harmonic functions as well as other elliptic equations, are similarly effective in treating geometric inequalities involving the ADM mass. In this paper, we survey recent results in this context, focusing on applications of spacetime harmonic functions to the asymptotically flat and asymptotically hyperbolic versions of the spacetime positive mass theorem, and addition…
For an infinite penny graph, we study the finite-dimensional property for the space of harmonic functions, or ancient solutions of the heat equation, of polynomial growth. We prove the asymptotically sharp dimensional estimate for the above spaces.
In this paper, we prove that associated with a sub-static asymptotically flat manifold endowed with a harmonic potential there is a one-parameter family $\{F_β\}$ of functions which are monotone along the level-set flow of the potential. Such monotonicity holds up to the optimal threshold $β=\frac{n-2}{n-1}$ and allows us to prove a geometric capacitary inequality where the capacity of the horizon plays the same role as the ADM mass in the celebrated Riemannian Penrose Inequality.
We obtain a volume growth and curvature decay result for various classes of complete, noncompact Riemannian metrics in dimension 4; in particular our method applies to anti-self-dual or Kahler metrics with zero scalar curvature, and metrics with harmonic curvature. Similar results are known for Einstein metrics, but our analysis differs from the Einstein case in that (1) we consider more generally a fourth order system in the metric, and (2) we do not assume any pointwise Ricci curvature bound.
We extend Bony's propagation of support argument \cite{Bony} to $C^1$ solutions of the non-homogeneous sub-elliptic $p-$Laplacian associated to a system of smooth vector fields satisfying Hörmander's finite rank condition. As a consequence we prove a strong maximum principle and strong comparison principle that generalize results of Tolksdorf \cite{Tolksdorf}.
