Nonnegative Ricci Curvature Research Articles

Liouville theorem for [formula omitted]-harmonic maps under non-negative [formula omitted]-Ricci curvature for non-positive [formula omitted

Let V be a C1-vector field on an n-dimensional complete Riemannian manifold (M,g). We prove a Liouville theorem for V-harmonic maps satisfying various growth conditions from complete Riemannian manifolds with non-negative (m,V)-Ricci curvature for m∈[−∞,0]∪[n,+∞] into Cartan–Hadamard manifolds, which extends Cheng’s Liouville theorem proved in Cheng (1980) for sublinear growth harmonic maps from complete Riemannian manifolds with non-negative Ricci curvature into Cartan–Hadamard manifolds. We also prove a Liouville theorem for V-harmonic maps from complete Riemannian manifolds with non-negative (m,V)-Ricci curvature for m∈[−∞,0]∪[n,+∞] into regular geodesic balls of Riemannian manifolds with positive upper sectional curvature bound, which extends the results of Hildebrandt et al. (1980) and Choi (1982). Our probabilistic proof of Liouville theorem for several growth V-harmonic maps into Hadamard manifolds enhances an incomplete argument in Stafford (1990). Our results extend the results due to Chen et al. (2012) and Qiu (2017) in the case of m=+∞ on the Liouville theorem for bounded V-harmonic maps from complete Riemannian manifolds with non-negative (∞,V)-Ricci curvature into regular geodesic balls of Riemannian manifolds with positive sectional curvature upper bound. Finally, we establish a connection between the Liouville property of V-harmonic maps and the recurrence property of ΔV-diffusion processes on manifolds. Our results are new even in the case V=∇f for f∈C2(M).

Optimal diameter estimate of three-dimensional Ricci limit spaces

In this note, we prove that positive scalar curvature can pass to three dimensional Ricci limit spaces of non-negative Ricci curvature when it splits off a line. As a corollary, we obtain an optimal Bonnet-Myers type upper bound. Moreover, we obtain a similar statement in all dimensions for Alexandrov spaces of non-negative curvature.

Some characterizations of Bach solitons via Ricci curvature

In this short note we provide some results for Bach solitons under different assumptions. In fact, under either non-negative or non-positive Ricci curvature condition we are able to show that a Bach soliton must be Bach-flat, since it satisfies a finite weighted Dirichlet integral condition or a parabolicity condition jointly with some regularity conditions L∞ or Lp on gradient of the potential function.

No breathers theorem for noncompact harmonic Ricci flows with asymptotically nonnegative Ricci curvature

Abstract By using the monotonicity of the log Sobolev functionals, we prove a no breathers theorem for noncompact harmonic Ricci flows under conditions on infimum of log Sobolev functionals and curvatures. As an application, we obtain a no breathers theorem for noncompact harmonic Ricci flows with asymptotically nonnegative Ricci curvature.

Uryson width of three dimensional mean convex domain with non-negative Ricci curvature

We prove that for every three dimensional manifold with nonnegative Ricci curvature and strictly mean convex boundary (non-empty), there exists a Morse function so that each connected component of its level sets has a uniform diameter bound, which depends only on the lower bound of mean curvature. This gives an upper bound of Uryson 1-width for those three manifolds with boundary.

Separation Conditions for Iterated Function Systems with Overlaps on Riemannian Manifolds

We formulate the weak separation condition and the finite type condition for conformal iterated function systems on Riemannian manifolds with nonnegative Ricci curvature, and generalize the main theorems by Lau et al. (Monatsch Math 156:325–355, 2009). We also obtain a formula for the Hausdorff dimension of a self-similar set defined by an iterated function system satisfying the finite type condition, generalizing a corresponding result by Jin and Yau (Commun Anal Geom 13:821–843, 2005) and Lau and Ngai (Adv Math 208:647–671, 2007) on Euclidean spaces. Moreover, we obtain a formula for the Hausdorff dimension of a graph self-similar set generated by a graph-directed iterated function system satisfying the graph finite type condition, extending a result by Ngai et al. (Nonlinearity 23:2333–2350, 2010).

Hodge theory on ALG∗ manifolds

Abstract We develop a Fredholm theory for the Hodge Laplacian in weighted spaces on ALG∗ manifolds in dimension four. We then give several applications of this theory. First, we show the existence of harmonic functions with prescribed asymptotics at infinity. A corollary of this is a non-existence result for ALG∗ manifolds with non-negative Ricci curvature having group Γ = { e } \Gamma=\{e\} at infinity. Next, we prove a Hodge decomposition for the first de Rham cohomology group of an ALG∗ manifold. A corollary of this is vanishing of the first Betti number for any ALG∗ manifold with non-negative Ricci curvature. Another application of our analysis is to determine the optimal order of ALG∗ gravitational instantons.

Critical metrics of the volume functional on three‐dimensional manifolds

AbstractIn this paper, we prove the three‐dimensional conjecture with nonnegative Ricci curvature. Moreover, we establish rigidity theorems for three‐dimensional compact, oriented, connected V‐static metrics with nonnegative Ricci curvature. Finally, we obtain classification results on three‐dimensional vacuum static space and Miao–Tam critical metric with nonnegative Ricci curvature.

Remark on an inequality for closed hypersurfaces in complete manifolds with nonnegative Ricci curvature

Remark on an inequality for closed hypersurfaces in complete manifolds with nonnegative Ricci curvature

Harmonic functions with polynomial growth on manifolds with nonnegative Ricci curvature

Harmonic functions with polynomial growth on manifolds with nonnegative Ricci curvature

On 3-manifolds with pointwise pinched nonnegative Ricci curvature

There is a conjecture that a complete Riemannian 3-manifold with bounded sectional curvature, and pointwise pinched nonnegative Ricci curvature, must be flat or compact. We show that this is true when the negative part (if any) of the sectional curvature decays quadratically.

Existence of [formula omitted]-dimension and entropy dimension of self-conformal measures on Riemannian manifolds

Peres and Solomyak proved that on Rn, the limits defining the Lq-dimension for any q∈(0,∞)∖{1}, and the entropy dimension of a self-conformal measure exist, without assuming any separation condition. By introducing the notions of heavy maximal packings and partitions, we prove that on a doubling metric space the Lq-dimension, q∈(0,∞)∖{1}, is equivalent to the generalized dimension. We also generalize the result on the existence of the Lq-dimension to self-conformal measures on complete Riemannian manifolds with the doubling property. In particular, these results hold for complete Riemannian manifolds with nonnegative Ricci curvature. Moreover, by assuming that the measure is doubling, we extend the result on the existence of the entropy dimension to self-conformal measures on complete Riemannian manifolds.

An overdetermined problem of the biharmonic operator on Riemannian manifolds

Let (M,g) be an n-dimensional complete Riemannian manifold with nonnegative Ricci curvature. In this paper, we consider an overdetermined problem of the biharmonic operator on a bounded smooth domain Ω in M. We deduce that the overdetermined problem has a solution only if Ω is isometric to a ball in mathbb{R}^{n}. Our method is based on using a P-function and the maximum principle argument. This result is a generalization of the overdetermined problem for the biharmonic equation in Euclidean space.

Talenti’s Comparison Theorem for Poisson Equation and Applications on Riemannian Manifold with Nonnegative Ricci Curvature

In this article, we prove Talenti’s comparison theorem for Poisson equation on complete noncompact Riemannian manifold with nonnegative Ricci curvature. Furthermore, we obtain the Faber–Krahn inequality for the first eigenvalue of Dirichlet Laplacian, $$L^1$$ - and $$L^\infty $$ -moment spectrum, especially Saint-Venant theorem for torsional rigidity and a reverse Hölder inequality for eigenfunctions of Dirichlet Laplacian.

Monotonicity Formulas for Harmonic Functions in textrm{RCD}(0,N) Spaces

We generalize to the textrm{RCD}(0,N) setting a family of monotonicity formulas by Colding and Minicozzi for positive harmonic functions in Riemannian manifolds with nonnegative Ricci curvature. Rigidity and almost rigidity statements are also proven, the second appearing to be new even in the smooth setting. Motivated by the recent work in Agostiniani et al. (Invent. Math. 222(3):1033–1101, 2020), we also introduce the notion of electrostatic potential in textrm{RCD} spaces, which also satisfies our monotonicity formulas. Our arguments are mainly based on new estimates for harmonic functions in textrm{RCD}(K,N) spaces and on a new functional version of the ‘(almost) outer volume cone implies (almost) outer metric cone’ theorem.

The construction of $$\epsilon $$-splitting map

For a geodesic ball with non-negative Ricci curvature and almost maximal volume, without using compactness argument, we construct an $$\epsilon $$ -splitting map on a concentric geodesic ball with uniformly small radius. There are two new technical points in our proof. The first one is the way of finding n directional points by induction and stratified almost Gou–Gu Theorem. The other one is the error estimates of projections, which guarantee the n directional points we find really determine n different directions.

The rigidity of sharp spectral gap in non-negatively curved spaces

We extend the celebrated rigidity of the sharp first spectral gap underRic≥0 to compact infinitesimally Hilbertian spaces with non-negative (weak, also called synthetic) Ricci curvature and bounded (synthetic) dimension i.e. to so-called compact RCD0,N spaces; this is a category of metric measure spaces which in particular includes (Ricci) non-negatively curved Riemannian manifolds, Alexandrov spaces, Ricci limit spaces, Bakry–Émery manifolds along with products, certain quotients and measured Gromov–Hausdorff limits of such spaces. In precise terms, we show in such spaces,λ1=π2diam2if and only if the space is one dimensional with a constant density function. We use new techniques mixing Sobolev theory and singular 1D-localization which might also be of independent interest. As a consequence of the rigidity in the singular setting, we also derive almost rigidity results.

On the $$L^p$$ Spectrum of the Dirac Operator

Our main goal in the present paper is to expand the known class of open manifolds over which the $L^2$-spectrum of a general Dirac operator and its square is maximal. To achieve this, we first find sufficient conditions on the manifold so that the $L^p$-spectrum of the Dirac operator and its square is independent of $p$ for $p\geq 1$. Using the $L^1$-spectrum, which is simpler to compute, we generalize the class of manifolds over which the $L^p$-spectrum of the Dirac operator is the real line for all $p$. We also show that by applying the generalized Weyl criterion, we can find large classes of manifolds with asymptotically nonnegative Ricci curvature, or which are asymptotically flat, such that the $L^2$-spectrum of a general Dirac operator and its square is maximal.

The asymptotic behaviour of p-capacitary potentials in asymptotically conical manifolds

We study the asymptotic behaviour of the p-capacitary potential and of the weak inverse mean curvature flow of a bounded set along the ends of an asymptotically conical Riemannian manifolds with asymptotically nonnegative Ricci curvature.

Comparison Results for Poisson Equation with Mixed Boundary Condition on Manifolds

In this article, we establish a $$L^1$$ estimate for solutions to Poisson equation with mixed boundary condition, on complete noncompact manifolds with nonnegative Ricci curvature and compact manifolds with positive Ricci curvature respectively. On Riemann surfaces we obtain a Talenti-type comparison. Our results generalize main theorems in Alvino et al. (J Math Pures Appl 9(152):251–261, 2021) to Riemannian setting, and Chen–Li’s result (Talenti’s comparison theorem for poisson equation and applications on Riemannian manifold with nonnegative Ricci curvature, 2021. arXiv:2104.05568 ) to the case of variable Robin parameter.