scholarly journals Eigenvalues of the drifted Laplacian on complete metric measure spaces

2016 ◽  
Vol 19 (01) ◽  
pp. 1650001 ◽  
Author(s):  
Xu Cheng ◽  
Detang Zhou
Keyword(s):  
Lower Bound ◽  
Measure Space ◽  
Logarithmic Sobolev Inequality ◽  
Ricci Soliton ◽  
Ricci Solitons ◽  
Dimensional Manifold ◽  
Metric Measure Spaces ◽  
Smooth Metric Measure Space ◽  
Measure Spaces ◽  
Multiplicity Formula

In this paper, first we study a complete smooth metric measure space [Formula: see text] with the ([Formula: see text])-Bakry–Émery Ricci curvature [Formula: see text] for some positive constant [Formula: see text]. It is known that the spectrum of the drifted Laplacian [Formula: see text] for [Formula: see text] is discrete and the first nonzero eigenvalue of [Formula: see text] has lower bound [Formula: see text]. We prove that if the lower bound [Formula: see text] is achieved with multiplicity [Formula: see text], then [Formula: see text], [Formula: see text] is isometric to [Formula: see text] for some complete [Formula: see text]-dimensional manifold [Formula: see text] and by passing an isometry, [Formula: see text] must split off a gradient shrinking Ricci soliton [Formula: see text], [Formula: see text]. This result can be applied to gradient shrinking Ricci solitons. Secondly, we study the drifted Laplacian [Formula: see text] for properly immersed self-shrinkers in the Euclidean space [Formula: see text], [Formula: see text] and show the discreteness of the spectrum of [Formula: see text] and a logarithmic Sobolev inequality.

HARMONIC MAPS FROM SMOOTH METRIC MEASURE SPACES

2012 ◽  
Vol 23 (09) ◽  
pp. 1250095 ◽  
Author(s):  
GUOFANG WANG ◽  
DELIANG XU
Keyword(s):  
Riemannian Manifold ◽  
Ricci Tensor ◽  
Measure Space ◽  
Harmonic Maps ◽  
Harmonic Map ◽  
Metric Measure Spaces ◽  
Rigidity Results ◽  
Smooth Metric Measure Space ◽  
Smooth Metric Measure Spaces ◽  
Measure Spaces

In this paper, we study a generalized harmonic map, ϕ-harmonic map, from a smooth metric measure space (M, g, e-ϕ dv) into a Riemannian manifold. We proved various rigidity results for the ϕ-harmonic maps under conditions in terms of the Bakry–Émery Ricci tensor.

scholarly journals Inscribed Radius Bounds for Lower Ricci Bounded Metric Measure Spaces with Mean Convex Boundary

2020 ◽  
Author(s):  
Annegret Burtscher ◽  
◽  
Christian Ketterer ◽  
Robert J. McCann ◽  
Eric Woolgar ◽  
...  
Keyword(s):  
Lower Bound ◽  
Mean Curvature ◽  
Measure Space ◽  
Riemannian Curvature ◽  
Metric Measure Spaces ◽  
Contraction Property ◽  
Measure Spaces ◽  
Generalized Mean Curvature ◽  
Measure Contraction Property ◽  
Mean Convex

Consider an essentially nonbranching metric measure space with the measure contraction property of Ohta and Sturm, or with a Ricci curvature lower bound in the sense of Lott, Sturm and Villani. We prove a sharp upper bound on the inscribed radius of any subset whose boundary has a suitably signed lower bound on its generalized mean curvature. This provides a nonsmooth analog to a result of Kasue (1983) and Li (2014). We prove a stability statement concerning such bounds and - in the Riemannian curvature-dimension (RCD) setting - characterize the cases of equality.

scholarly journals ε-Regularity and Structure of Four-dimensional Shrinking Ricci Solitons

2018 ◽  
Vol 2020 (5) ◽  
pp. 1511-1574 ◽  
Author(s):  
Shaosai Huang
Keyword(s):  
Lower Bound ◽  
Ricci Soliton ◽  
Detailed Account ◽  
Ricci Solitons ◽  
Dimensional Manifold ◽  
Regularity Theorem ◽  
Bounded Curvature ◽  
Soliton Metric

Abstract A closed four-dimensional manifold cannot possess a non-flat Ricci soliton metric with arbitrarily small $L^{2}$-norm of the curvature. In this paper, we localize this fact in the case of gradient shrinking Ricci solitons by proving an $\varepsilon $-regularity theorem, thus confirming a conjecture of Cheeger–Tian [20]. As applications, we will also derive structural results concerning the degeneration of the metrics on a family of complete non-compact four-dimensional gradient shrinking Ricci solitons without a uniform entropy lower bound. In the appendix, we provide a detailed account of the equivariant good chopping theorem when collapsing with locally bounded curvature happens.

scholarly journals Distribution-Valued Ricci Bounds for Metric Measure Spaces, Singular Time Changes, and Gradient Estimates for Neumann Heat Flows

2020 ◽  
Vol 30 (6) ◽  
pp. 1648-1711
Author(s):  
Karl-Theodor Sturm
Keyword(s):  
Lower Bound ◽  
Second Fundamental Form ◽  
Gradient Estimates ◽  
List Type ◽  
Metric Measure Spaces ◽  
Heat Flows ◽  
Measure Spaces ◽  
Neumann Laplacian ◽  
Boundary Curvature ◽  
Time Changes

AbstractWe will study metric measure spaces $$(X,\mathsf{d},{\mathfrak {m}})$$ ( X , d , m ) beyond the scope of spaces with synthetic lower Ricci bounds. In particular, we introduce distribution-valued lower Ricci bounds $$\mathsf{BE}_1(\kappa ,\infty )$$ BE 1 ( κ , ∞ ) for which we prove the equivalence with sharp gradient estimates, the class of which will be preserved under time changes with arbitrary $$\psi \in \mathrm {Lip}_b(X)$$ ψ ∈ Lip b ( X ) , and which are satisfied for the Neumann Laplacian on arbitrary semi-convex subsets $$Y\subset X$$ Y ⊂ X . In the latter case, the distribution-valued Ricci bound will be given by the signed measure $$\kappa = k\,{\mathfrak {m}}_Y + \ell \,\sigma _{\partial Y}$$ κ = k m Y + ℓ σ ∂ Y where k denotes a variable synthetic lower bound for the Ricci curvature of X and $$\ell $$ ℓ denotes a lower bound for the “curvature of the boundary” of Y, defined in purely metric terms. We also present a new localization argument which allows us to pass on the RCD property to arbitrary open subsets of RCD spaces. And we introduce new synthetic notions for boundary curvature, second fundamental form, and boundary measure for subsets of RCD spaces.

scholarly journals The p-Royden and p-Harmonic Boundaries for Metric Measure Spaces

2015 ◽  
Vol 3 (1) ◽  
Author(s):  
Marcello Lucia ◽  
Michael J. Puls
Keyword(s):  
Real Number ◽  
Measure Space ◽  
Metric Measure Space ◽  
Metric Measure Spaces ◽  
Type Problem ◽  
Locally Compact ◽  
Dirichlet Type ◽  
Algebra Of Functions ◽  
Measure Spaces ◽  
Dirichlet Type Problem

Abstract Let p be a real number greater than one and let X be a locally compact, noncompact metric measure space that satisfies certain conditions. The p-Royden and p-harmonic boundaries of X are constructed by using the p-Royden algebra of functions on X and a Dirichlet type problem is solved for the p-Royden boundary. We also characterize the metric measure spaces whose p-harmonic boundary is empty.

GENERALIZED MORREY SPACES OVER NONHOMOGENEOUS METRIC MEASURE SPACES

2016 ◽  
Vol 103 (2) ◽  
pp. 268-278 ◽  
Author(s):  
GUANGHUI LU ◽  
SHUANGPING TAO
Keyword(s):  
Integral Operator ◽  
Measure Space ◽  
Fractional Integral ◽  
Morrey Spaces ◽  
Marcinkiewicz Integral ◽  
Metric Measure Spaces ◽  
Generalized Morrey Spaces ◽  
Measure Spaces ◽  
Definition Of ◽  
Marcinkiewicz Integral Operator

Let $({\mathcal{X}},d,\unicode[STIX]{x1D707})$ be a nonhomogeneous metric measure space satisfying the so-called upper doubling and the geometric doubling conditions. In this paper, the authors give the natural definition of the generalized Morrey spaces on $({\mathcal{X}},d,\unicode[STIX]{x1D707})$, and then investigate some properties of the maximal operator, the fractional integral operator and its commutator, and the Marcinkiewicz integral operator.

scholarly journals Boundedness of Commutators of Marcinkiewicz Integrals on Nonhomogeneous Metric Measure Spaces

2015 ◽  
Vol 2015 ◽  
pp. 1-12 ◽  
Author(s):  
Guanghui Lu ◽  
Shuangping Tao
Keyword(s):  
Hardy Spaces ◽  
Measure Space ◽  
Morrey Spaces ◽  
Marcinkiewicz Integral ◽  
Metric Measure Spaces ◽  
Doubling Condition ◽  
Lebesgue Spaces ◽  
Marcinkiewicz Integrals ◽  
Measure Spaces ◽  
Weak Lebesgue Spaces

Let(X,d,μ)be a metric measure space satisfying the upper doubling condition and geometrically doubling condition in the sense of Hytönen. The aim of this paper is to establish the boundedness of commutatorMbgenerated by the Marcinkiewicz integralMand Lipschitz functionb. The authors prove thatMbis bounded from the Lebesgue spacesLp(μ)to weak Lebesgue spacesLq(μ)for1≤p<n/β, from the Lebesgue spacesLp(μ)to the spacesRBMO(μ)forp=n/β, and from the Lebesgue spacesLp(μ)to the Lipschitz spacesLip(β-n/p)(μ)forn/β<p≤∞. Moreover, some results in Morrey spaces and Hardy spaces are also discussed.

scholarly journals Boundedness of Lusin-area andgλ*functions on localized Morrey-Campanato spaces over doubling metric measure spaces

2011 ◽  
Vol 9 (3) ◽  
pp. 245-282 ◽  
Author(s):  
Haibo Lin ◽  
Eiichi Nakai ◽  
Dachun Yang
Keyword(s):  
Lebesgue Measure ◽  
Measure Space ◽  
Admissible Function ◽  
Regularity Property ◽  
Metric Measure Space ◽  
Metric Measure Spaces ◽  
Area Function ◽  
Measure Spaces ◽  
Lusin Area Function ◽  
Dimensional Lebesgue Measure

Letχbe a doubling metric measure space andρan admissible function onχ. In this paper, the authors establish some equivalent characterizations for the localized Morrey-Campanato spacesερα,p(χ)and Morrey-Campanato-BLO spacesε̃ρα,p(χ)whenα∈(-∞,0)andp∈[1,∞). Ifχhas the volume regularity Property(P), the authors then establish the boundedness of the Lusin-area function, which is defined via kernels modeled on the semigroup generated by the Schrödinger operator, fromερa,p(χ)toε̃ρa,p(χ)without invoking any regularity of considered kernels. The same is true for thegλ*function and, unlike the Lusin-area function, in this case,χis even not necessary to have Property(P). These results are also new even forℝdwith thed-dimensional Lebesgue measure and have a wide applications.

scholarly journals Fractional Maximal Functions in Metric Measure Spaces

2013 ◽  
Vol 1 ◽  
pp. 147-162 ◽  
Author(s):  
Toni Heikkinen ◽  
Juha Lehrbäck ◽  
Juho Nuutinen ◽  
Heli Tuominen
Keyword(s):  
Maximal Function ◽  
Lipschitz Function ◽  
Measure Space ◽  
Continuous Functions ◽  
Maximal Operators ◽  
Metric Measure Spaces ◽  
Hölder Continuous ◽  
Fractional Maximal Function ◽  
Measure Spaces ◽  
Hölder Continuous Functions

Abstract We study the mapping properties of fractional maximal operators in Sobolev and Campanato spaces in metric measure spaces. We show that, under certain restrictions on the underlying metric measure space, fractional maximal operators improve the Sobolev regularity of functions and map functions in Campanato spaces to Hölder continuous functions. We also give an example of a space where fractional maximal function of a Lipschitz function fails to be continuous.

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