Harmonic functions on asymptotic cones with Euclidean volume growth

Honda, Shinya

doi:10.2969/jmsj/06710069

Cited by 12 publications

(13 citation statements)

References 64 publications

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“…In particular, tX = ∂B t (o). With covering technique and Bishop-Gromov volume comparison, the cone CX implies the co-area formula (see Proposition 7.6 in [26] for instance) (6.5)…”

Section: Limiting Cones From Minimal Hypersurfacesmentioning

confidence: 99%

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Minimal hypersurfaces in manifolds of Ricci curvature bounded below

Ding¹

2021

Preprint

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In this paper, we study the angle estimate of distance functions from minimal hypersurfaces in manifolds of Ricci curvature bounded from below using Colding's method in [13]. With Cheeger-Colding theory, we obtain the Laplacian comparison for limits of distance functions from minimal hypersurfaces in the version of Ricci limit space. As an application, if a sequence of minimal hypersurfaces converges to a metric coneobtained from ambient manifolds of almost nonnegative Ricci curvature, then we can prove a Frankel property for the cross section Y of CY . Namely, Y has only one connected component in X.

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“…In particular, tX = ∂B t (o). With covering technique and Bishop-Gromov volume comparison, the cone CX implies the co-area formula (see Proposition 7.6 in [26] for instance) (6.5)…”

Section: Limiting Cones From Minimal Hypersurfacesmentioning

confidence: 99%

“…For any point x ∈ X and r, t > 0, let B r (tx) denote the metric ball in tX = ∂B t (o) with the radius r and centered at tx. With Bishop-Gromov volume comparison, there is a constant c k ≥ 1 depending only on k such that (see (8.4) in the Appendix II or Proposition 7.9 in [26] by Honda for instance)…”

Section: Limiting Cones From Minimal Hypersurfacesmentioning

confidence: 99%

Minimal hypersurfaces in manifolds of Ricci curvature bounded below

Ding¹

2021

Preprint

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“…for any K, Ω ⊂ C. From the co-area formula for non-collapsed metric cones (see Proposition 7.6 in [42] by Honda for instance), for any t > s > 0 and any δ > 0 we have (4.28)…”

Section: Tangent Cones Of Limits Of Minimal Graphs Over Manifoldsmentioning

confidence: 99%

Poincaré inequality on minimal graphs over manifolds and applications

Ding¹

2021

Preprint

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Let B2(p) be an n-dimensional smooth geodesic ball with Ricci curvature ≥ −(n − 1)κ 2 for some κ ≥ 0. We establish the Sobolev inequality and the uniform Neumann-Poincaré inequality on each minimal graph over B1(p) by combining Cheeger-Colding theory and the current theory from geometric measure theory, where the constants in the inequalities only depends on n, κ, the lower bound of the volume of B1(p). As applications, we derive gradient estimates and a Liouville theorem for a minimal graph M over a smooth complete noncompact manifold Σ of nonnegative Ricci curvature and Euclidean volume growth. Furthermore, we can show that any tangent cone of Σ at infinity splits off a line isometrically provided the graphic function of M admits linear growth.

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“…And

u_{i}

is harmonic on

B_{p} false(R_{i} false) = B_{i} false(1 false)

satisfying the following property:

\begin{matrix} true \\ right \lim_{i \to \infty} {false| u_{i} \circ Ψ_{\infty, i} - u_{\infty} false|}_{L^{\infty} ((), B_{\infty}, false(1 false))} = 0, u_{i} (p) = 0, \end{matrix}

where

{normalΨ}_{\infty, i} : B_{\infty} false(1 false) \to B_{i} false(1 false)

is an

ε_{i}

‐Gromov–Hausdorff approximation, and

\lim_{i \to \infty} ε_{i} = 0

. From [18, Proposition 3.4], we have

\begin{matrix} true \\ right \lim_{i \to \infty} I_{u_{i}} (t) = I_{u_{\infty}} (t), \forall t \in (0, 1] . \end{matrix}

Step (2). Let

d = α_{1} + 1

, we will prove that there exists

i_{0} = i_{0} > 0

such that if

i ⩾ i_{0}

, then

I_{u_{i}} false(r false) ⩽ 2^{2 d} \cdot I_{u_{i}} (\frac{r}{2}), \forall r \in false[4^{<...>}

…”

Section: Frequency and Existence Of Harmonic Functionmentioning

confidence: 99%

The growth rate of harmonic functions

2020

J. London Math. Soc.

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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 non-negative curvature. On complete Riemannian manifolds with non-negative Ricci curvature and maximal volume growth, further assume the dimension of the manifold is not less than three, we prove that quantitative strong unique continuation yields the existence of non-constant 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.

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Harmonic functions on asymptotic cones with Euclidean volume growth

Cited by 12 publications

References 64 publications

Minimal hypersurfaces in manifolds of Ricci curvature bounded below

Minimal hypersurfaces in manifolds of Ricci curvature bounded below

Poincaré inequality on minimal graphs over manifolds and applications

The growth rate of harmonic functions

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