The growth rate of harmonic functions
5
Citation
32
Reference
10
Related Paper
Citation Trend
Abstract:
We study the growth rate of harmonic functions in two aspects: gradient estimate and frequency. We obtain the sharp gradient estimate of positive harmonic function in geodesic ball of complete surface with nonnegative curvature. On complete Riemannian manifolds with nonnegative Ricci curvature and maximal volume growth, further assume the dimension of the manifold is not less than 3, we prove that quantitative strong unique continuation yields the existence of nonconstant polynomial growth harmonic functions. Also the uniform bound of frequency for linear growth harmonic functions on such manifolds is obtained, and this confirms a special case of Colding–Minicozzi's conjecture on frequency.Keywords:
Riemannian manifold
Harmonic
Ball (mathematics)
Manifold (fluid mechanics)
A consequence of the surgery theorem of Gromov and Lawson is that every closed, simply-connected 6-manifold admits a Riemannian metric of positive scalar curvature. For metrics of positive Ricci curvature it is widely open whether a similar result holds; there are no obstructions known for those manifolds to admit a metric of positive Ricci curvature, while the number of examples known is limited. In this article we introduce a new description of certain $6k$-dimensional manifolds via labeled bipartite graphs and use an earlier result of the author to construct metrics of positive Ricci curvature on these manifolds. In this way we obtain many new examples, both spin and non-spin, of $6k$-dimensional manifolds with a metric of positive Ricci curvature.
Ricci Flow
Manifold (fluid mechanics)
Yamabe flow
Cite
Citations (1)
It is well known there is no non-constant harmonic map from a closed Riemannian manifold of positive Ricci curvature to a complete Riemannian manifold with non-positive sectional curvature. If one reduces the assumption on the Ricci curvature to one on th
Manifold (fluid mechanics)
Riemannian manifold
Cite
Citations (1)
It is well known that no non-trivial Killing vector field exists on a compact Riemannian manifold of negative Ricci curvature; analogously, no non-trivial harmonic one-form exists on a compact manifold of positive Ricci curvature. One can consider the following, more general, problem. By reducing the assumption on the Ricci curvature to one on the scalar curvature, such vanishing theorems cannot hold in general. This raises the question: “What information can we obtain from the existence of non-trivial Killing vector fields (or, respectively, harmonic one-forms)?” This paper gives answers to this problem; the results obtained are optimal. 2000 Mathematics Subject Classification 53C20 (primary), 53C24 (secondary).
Riemannian manifold
Killing vector field
Manifold (fluid mechanics)
Cite
Citations (5)
Let (M, g) be an n-dimensional Riemannian manifold. We say M has k-positive Ricci curvature if at each point p ∈ M the sum of the k smallest eigenvalues of the Ricci curvature at p is positive. We say that the k-positive Ricci curvature is bounded below by α if the sum of the k smallest eigenvalues is greater than α. Note that n-positive Ricci curvature is equivalent to positive scalar curvature and one-positive Ricci curvature is equivalent to positive Ricci curvature. We first describe some basic connect sum and surgery results for k-positive Ricci curvature that are direct generalizations of the well known results for positive scalar curvature (n-positive Ricci curvature). Using these results we construct examples that motivate questions and conjectures in the cases of 2-positive and (n − 1)-positive Ricci curvature. In particular:
Ricci Flow
Cite
Citations (7)
In Riemannian geometry the prescribed Ricci curvature problem is as follows: given a smooth manifold $M$ and a symmetric 2-tensor $r$, construct a metric on $M$ whose Ricci tensor equals $r$. In particular, DeTurck and Koiso proved the following celebrated result: the Ricci curvature uniquely determines the Levi-Civita connection on any compact Einstein manifold with non-negative section curvature. In the present paper we generalize the result of DeTurck and Koiso for a Riemannian manifold with non-negative section curvature. In addition, we extended our result to complete non-compact Riemannian manifolds with nonnegative sectional curvature and with finite total scalar curvature.
Metric tensor
Riemannian manifold
Manifold (fluid mechanics)
Cite
Citations (0)
In this paper, we obtain the Ricci curvature and the scalar curvature on SU(3) with some left invariant Riemannian metric. And then we get a necessary and sufficient condition for the scalar curvature (resp. the Ricci curvature) on the Riemannian manifold SU(3) to be positive.
Ricci Flow
Yamabe flow
Cite
Citations (1)
Let N~(n+p) be a(n+p)-dimensional locally symmetric complete Riemannian manifold with sectional curvature K_N such that 12δ≤K_N≤1,Let M be a n-dimensional compact minimal submanifold in(N~(n+p).)A pinching problem with respect to the Ricci curvature of M is discussed.
Riemannian manifold
Manifold (fluid mechanics)
Cite
Citations (0)
Abstract Let M be an n -dimensional complete Riemannian manifold with Ricci curvature bounded from below. Let be an N -dimensional ( N < n ) complete, simply connected Riemannian manifold with nonpositive sectional curvature. We shall prove in this note that if there exists an isometric immersion φ of M into with the property that the immersed manifold is contained in a ball of radius R and that the mean curvature vector H of the immersion has bounded norm ∥ H ∥ > H 0 , ( H 0 > 0) then R > H −1 0 .
Riemannian manifold
Immersion
Ball (mathematics)
Minimal volume
Cite
Citations (8)
In this paper, we deal with a critical point metric of the total scalar curvature on a compact manifold $M$ . We prove that if the critical point metric has parallel Ricci tensor, then the manifold is isometric to a standard sphere. Moreover, we show that if an $n$ -dimensional Riemannian manifold is a warped product, or has harmonic curvature with non-parallel Ricci tensor, then it cannot be a critical point metric.
Critical point (mathematics)
Manifold (fluid mechanics)
Riemannian manifold
Metric tensor
Cite
Citations (19)
aSuppose that (M, g) is a complete Riemannian manifold with Ricci curvature bounded below by -K ) is a complete Riemannian manifold with sectional curvature bounded above by a constant > 0. Let u : is a heat equation for harmonic map. We estimate the energy density of u.
Harmonic map
Riemannian manifold
Manifold (fluid mechanics)
Heat equation
Constant (computer programming)
Harmonic
Constant curvature
Cite
Citations (0)