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Journal articles on the topic 'Ricci lower bound'

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Author: Grafiati

Published: 9 March 2023

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1

ZHANG, ZHOU. "RICCI LOWER BOUND FOR KÄHLER–RICCI FLOW." Communications in Contemporary Mathematics 16, no. 02 (April 2014): 1350053. http://dx.doi.org/10.1142/s0219199713500533.

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We provide general discussion on the lower bound of Ricci curvature along Kähler–Ricci flows over closed manifolds. The main result is the non-existence of Ricci lower bound for flows with finite time singularities and non-collapsed global volume. As an application, we give examples showing that positivity of Ricci curvature would not be preserved by Ricci flow in general.

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2

Hu, Zisheng, Yadong Jin, and Senlin Xu. "A Volume Comparison Estimate with Radially Symmetric Ricci Curvature Lower Bound and Its Applications." International Journal of Mathematics and Mathematical Sciences 2010 (2010): 1–14. http://dx.doi.org/10.1155/2010/758531.

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We extend the classical Bishop-Gromov volume comparison from constant Ricci curvature lower bound to radially symmetric Ricci curvature lower bound, and apply it to investigate the volume growth, total Betti number, and finite topological type of manifolds with nonasymptotically almost nonnegative Ricci curvature.

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3

Wei, Guofang. "Manifolds with a lower Ricci Curvature Bound." Surveys in Differential Geometry 11, no. 1 (2006): 203–28. http://dx.doi.org/10.4310/sdg.2006.v11.n1.a7.

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4

Xu, Guoyi. "Lower bound of Ricci flow's existence time." Bulletin of the London Mathematical Society 47, no. 5 (September 18, 2015): 759–70. http://dx.doi.org/10.1112/blms/bdv054.

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5

Liu, Gang. "Kähler manifolds with Ricci curvature lower bound." Asian Journal of Mathematics 18, no. 1 (2014): 69–100. http://dx.doi.org/10.4310/ajm.2014.v18.n1.a4.

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6

Bessières, L., G. Besson, G. Courtois, and S. Gallot. "Differentiable rigidity under Ricci curvature lower bound." Duke Mathematical Journal 161, no. 1 (January 2012): 29–67. http://dx.doi.org/10.1215/00127094-1507272.

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7

LING, JUN. "A LOWER BOUND OF THE FIRST DIRICHLET EIGENVALUE OF A COMPACT MANIFOLD WITH POSITIVE RICCI CURVATURE." International Journal of Mathematics 17, no. 05 (May 2006): 605–17. http://dx.doi.org/10.1142/s0129167x06003631.

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We give a lower bound for the first Dirichlet eigenvalue for a compact manifold with positive Ricci curvature in terms of the lower bound of the Ricci curvature and the interior radius. The result sharpens earlier estimates.

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8

Sturm, Karl-Theodor. "Distribution-Valued Ricci Bounds for Metric Measure Spaces, Singular Time Changes, and Gradient Estimates for Neumann Heat Flows." Geometric and Functional Analysis 30, no. 6 (November 20, 2020): 1648–711. http://dx.doi.org/10.1007/s00039-020-00554-0.

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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.

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9

Wang, Yu, and Xiangwen Zhang. "Measure Estimates, Harnack Inequalities and Ricci Lower Bound." Acta Mathematica 55 (November 2018): 21–51. http://dx.doi.org/10.4467/20843828am.18.002.9718.

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10

Erdoğan, Mehmet. "A lower bound estimation for the Ricci curvature of a hypersurface in a hyperbolic space." Studia Scientiarum Mathematicarum Hungarica 46, no. 4 (December 1, 2009): 539–46. http://dx.doi.org/10.1556/sscmath.46.2009.4.1106.

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In some previous papers the author gave an upper bound estimation for the Ricci curvature of a hypersurface in a hyperbolic space and in a sphere, see [4] and [5]. In the present paper, we give a lower bound estimation for the Ricci curvature of a compact connected embedded hypersurface in a hyperbolic space via the maximum principle given by H. Omori in [11].

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11

Sakurai, Yohei. "Comparison Geometry of Manifolds with Boundary under a Lower Weighted Ricci Curvature Bound." Canadian Journal of Mathematics 72, no. 1 (October 24, 2018): 243–80. http://dx.doi.org/10.4153/s0008414x1800007x.

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AbstractWe study Riemannian manifolds with boundary under a lower weighted Ricci curvature bound. We consider a curvature condition in which the weighted Ricci curvature is bounded from below by the density function. Under the curvature condition and a suitable condition for the weighted mean curvature for the boundary, we obtain various comparison geometric results.

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12

Deshmukh, Sharief, Hana Alsodais, and Nasser Bin Turki. "Some Results on Ricci Almost Solitons." Symmetry 13, no. 3 (March 7, 2021): 430. http://dx.doi.org/10.3390/sym13030430.

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We find three necessary and sufficient conditions for an n-dimensional compact Ricci almost soliton (M,g,w,σ) to be a trivial Ricci soliton under the assumption that the soliton vector field w is a geodesic vector field (a vector field with integral curves geodesics). The first result uses condition r2≤nσr on a nonzero scalar curvature r; the second result uses the condition that the soliton vector field w is an eigen vector of the Ricci operator with constant eigenvalue λ satisfying n2λ2≥r2; the third result uses a suitable lower bound on the Ricci curvature S(w,w). Finally, we show that an n-dimensional connected Ricci almost soliton (M,g,w,σ) with soliton vector field w is a geodesic vector field with a trivial Ricci soliton, if and only if, nσ−r is a constant along integral curves of w and the Ricci curvature S(w,w) has a suitable lower bound.

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13

Honda, Shouhei. "On low-dimensional Ricci limit spaces." Nagoya Mathematical Journal 209 (March 2013): 1–22. http://dx.doi.org/10.1017/s0027763000010667.

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AbstractWe call a Gromov–Hausdorff limit of complete Riemannian manifolds with a lower bound of Ricci curvature a Ricci limit space. Furthermore, we prove that any Ricci limit space has integral Hausdorff dimension, provided that its Hausdorff dimension is not greater than 2. We also classify 1-dimensional Ricci limit spaces.

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14

Dai, Xianzhe, and Guofang Wei. "A heat kernel lower bound for integral Ricci curvature." Michigan Mathematical Journal 52, no. 1 (April 2004): 61–69. http://dx.doi.org/10.1307/mmj/1080837734.

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15

Lee, Paul W. Y. "On Measure Contraction Property without Ricci Curvature Lower Bound." Potential Analysis 44, no. 1 (August 6, 2015): 27–41. http://dx.doi.org/10.1007/s11118-015-9496-z.

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16

Kim, Ju Seon, and Sang Og Kim. "An estimate of sectional curvatures of hypersurfaces with positive Ricci curvatures." Proceedings of the Edinburgh Mathematical Society 38, no. 1 (February 1995): 167–70. http://dx.doi.org/10.1017/s0013091500006283.

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Let M be a hypersurface in Euclidean space and let the Ricci curvature of M be bounded below by some nonnegative constant. In this paper, we estimate the sectional curvature of M in terms of the lower bound of Ricci curvature and the upper bound of mean curvature.

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17

Carbonaro, A., and G. Mauceri. "A note on bounded variation and heat semigroup on Riemannian manifolds." Bulletin of the Australian Mathematical Society 76, no. 1 (August 2007): 155–60. http://dx.doi.org/10.1017/s000497270003954x.

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In a recent paper Miranda Jr., Pallara, Paronetto and Preunkert have shown that the classical De Giorgi's heat kernel characterisation of functions of bounded variation on Euclidean space extends to Riemannian manifolds with Ricci curvature bounded from below and which satisfy a uniform lower bound estimate on the volume of geodesic balls of fixed radius. We give a shorter proof of the same result assuming only the lower bound on the Ricci curvature.

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18

Deshmukh, Sharief. "Almost Ricci solitons isometric to spheres." International Journal of Geometric Methods in Modern Physics 16, no. 05 (May 2019): 1950073. http://dx.doi.org/10.1142/s0219887819500737.

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We find a characterization of a sphere using a compact gradient almost Ricci soliton and the lower bound on the integral of Ricci curvature in the direction of potential field. Also, we use Poisson equation on a compact gradient almost Ricci soliton to find a characterization of the unit sphere.

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19

Zhu, Jonathan J. "Minimal hypersurfaces with small first eigenvalue in manifolds of positive Ricci curvature." Journal of Topology and Analysis 09, no. 03 (July 28, 2016): 505–32. http://dx.doi.org/10.1142/s1793525317500200.

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In this paper we exhibit deformations of the hemisphere [Formula: see text], [Formula: see text], for which the ambient Ricci curvature lower bound [Formula: see text] and the minimality of the boundary are preserved, but the first Laplace eigenvalue of the boundary decreases. The existence of these metrics suggests that any resolution of Yau’s conjecture on the first eigenvalue of minimal hypersurfaces in spheres would likely need to consider more geometric data than a Ricci curvature lower bound.

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20

Kalka, Morris, Elizabeth Mann, Dagang Yang, and Aleksey Zinger. "The Exponential Decay Rate of the Lower Bound for the First Eigenvalue of Compact Manifolds." International Journal of Mathematics 08, no. 03 (May 1997): 345–55. http://dx.doi.org/10.1142/s0129167x97000160.

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This paper provides the optimal exponential decay rate of the lower bound for the first positive eigenvalue of the Laplacian operator on a compact Riemannian manifold with a negative lower bound on the Ricci curvature and with large diameter. For manifolds with boundary, suitable convexity conditions are assumed.

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21

Blaga, Adara. "Geometric solitons in a D-homothetically deformed Kenmotsu manifold." Filomat 36, no. 1 (2022): 175–86. http://dx.doi.org/10.2298/fil2201175b.

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We consider almost Riemann and almost Ricci solitons in a D-homothetically deformed Kenmotsu manifold having as potential vector field a gradient vector field, a solenoidal vector field or the Reeb vector field of the deformed structure, and explicitly obtain the Ricci and scalar curvatures for some cases. We also provide a lower bound for the Ricci curvature of the initial Kenmotsu manifold when the deformed manifold admits a gradient almost Riemann or almost Ricci soliton.

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22

Veeravalli, Alain R. "A sharp lower bound for the Ricci curvature of bounded hypersurfaces in space forms." Bulletin of the Australian Mathematical Society 62, no. 1 (August 2000): 165–70. http://dx.doi.org/10.1017/s0004972700018591.

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23

Hassannezhad, Asma, Gerasim Kokarev, and Iosif Polterovich. "Eigenvalue inequalities on Riemannian manifolds with a lower Ricci curvature bound." Journal of Spectral Theory 6, no. 4 (2016): 807–35. http://dx.doi.org/10.4171/jst/143.

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24

Chu, Jianchun, and Wenshuai Jiang. "A note on nodal sets on manifolds with lower Ricci bound." Methods and Applications of Analysis 27, no. 4 (2020): 359–74. http://dx.doi.org/10.4310/maa.2020.v27.n4.a3.

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25

Han, Bang-Xian. "Measure rigidity of synthetic lower Ricci curvature bound on Riemannian manifolds." Advances in Mathematics 373 (October 2020): 107327. http://dx.doi.org/10.1016/j.aim.2020.107327.

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26

Chen, Lina, Xiaochun Rong, and Shicheng Xu. "Quantitative volume space form rigidity under lower Ricci curvature bound I." Journal of Differential Geometry 113, no. 2 (October 2019): 227–72. http://dx.doi.org/10.4310/jdg/1571882427.

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27

Chen, Lina, Xiaochun Rong, and Shicheng Xu. "Quantitative volume space form rigidity under lower Ricci curvature bound II." Transactions of the American Mathematical Society 370, no. 6 (November 16, 2017): 4509–23. http://dx.doi.org/10.1090/tran/7279.

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28

Inahama, Yuzuru. "Explicit lower bound of the Ricci tensor on free loop algebras." Journal of Mathematics of Kyoto University 42, no. 3 (2002): 465–75. http://dx.doi.org/10.1215/kjm/1250283844.

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29

Shen, Liangming. "Smooth approximation of conic Kähler metric with lower Ricci curvature bound." Pacific Journal of Mathematics 284, no. 2 (August 30, 2016): 455–74. http://dx.doi.org/10.2140/pjm.2016.284.455.

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30

Colding, Tobias H., and William P. Minicozzi. "On Function Theory on Spaces with a Lower Ricci Curvature Bound." Mathematical Research Letters 3, no. 2 (1996): 241–46. http://dx.doi.org/10.4310/mrl.1996.v3.n2.a9.

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31

Tadano, Homare. "Remark on a lower diameter bound for compact shrinking Ricci solitons." Differential Geometry and its Applications 66 (October 2019): 231–41. http://dx.doi.org/10.1016/j.difgeo.2019.06.005.

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32

Munteanu, Ovidiu, and Jiaping Wang. "Geometry of shrinking Ricci solitons." Compositio Mathematica 151, no. 12 (July 29, 2015): 2273–300. http://dx.doi.org/10.1112/s0010437x15007496.

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The main purpose of this paper is to investigate the curvature behavior of four-dimensional shrinking gradient Ricci solitons. For such a soliton $M$ with bounded scalar curvature $S$, it is shown that the curvature operator $\text{Rm}$ of $M$ satisfies the estimate $|\text{Rm}|\leqslant cS$ for some constant $c$. Moreover, the curvature operator $\text{Rm}$ is asymptotically nonnegative at infinity and admits a lower bound $\text{Rm}\geqslant -c(\ln (r+1))^{-1/4}$, where $r$ is the distance function to a fixed point in $M$. As an application, we prove that if the scalar curvature converges to zero at infinity, then the soliton must be asymptotically conical. As a separate issue, a diameter upper bound for compact shrinking gradient Ricci solitons of arbitrary dimension is derived in terms of the injectivity radius.

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33

Cheng, Xu, and Detang Zhou. "Eigenvalues of the drifted Laplacian on complete metric measure spaces." Communications in Contemporary Mathematics 19, no. 01 (November 24, 2016): 1650001. http://dx.doi.org/10.1142/s0219199716500012.

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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.

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34

Chang, Jeongwook. "EVOLUTION EQUATIONS ON A RIEMANNIAN MANIFOLD WITH A LOWER RICCI CURVATURE BOUND." East Asian mathematical journal 30, no. 1 (January 31, 2014): 79–91. http://dx.doi.org/10.7858/eamj.2014.008.

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35

Song, Jian, and Xiaowei Wang. "The greatest Ricci lower bound, conical Einstein metrics and Chern number inequality." Geometry & Topology 20, no. 1 (February 29, 2016): 49–102. http://dx.doi.org/10.2140/gt.2016.20.49.

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36

Liu, Gang. "On the tangent cone of Kähler manifolds with Ricci curvature lower bound." Mathematische Annalen 370, no. 1-2 (March 27, 2017): 649–67. http://dx.doi.org/10.1007/s00208-017-1536-0.

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37

Hineva, Stefka. "Submanifolds for which a lower bound of the Ricci curvature is achieved." Journal of Geometry 88, no. 1-2 (March 2008): 53–69. http://dx.doi.org/10.1007/s00022-007-1920-0.

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38

GULER, DINCER, and FANGYANG ZHENG. "ON RICCI RANK OF CARTAN–HADAMARD MANIFOLDS." International Journal of Mathematics 13, no. 06 (August 2002): 557–78. http://dx.doi.org/10.1142/s0129167x02001459.

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In this article, we prove that the maximum rank r of the Ricci tensor of a Cartan–Hadamard manifold Mn satisfies the inequality 2r - 1 ≥ n - s, where n is the dimension and s is the core number, which measures the flatness of Mn. Examples show that this lower bound is sharp.

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39

Huang, Shaosai. "ε-Regularity and Structure of Four-dimensional Shrinking Ricci Solitons." International Mathematics Research Notices 2020, no. 5 (April 18, 2018): 1511–74. http://dx.doi.org/10.1093/imrn/rny069.

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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.

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40

Blaga, Adara. "On warped product gradient η-Ricci solitons." Filomat 31, no. 18 (2017): 5791–801. http://dx.doi.org/10.2298/fil1718791b.

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If the potential vector field of an ?-Ricci soliton is of gradient type, using Bochner formula, we derive from the soliton equation a nonlinear second order PDE. In a particular case of irrotational potential vector field we prove that the soliton is completely determined by f . We give a way to construct a gradient ?-Ricci soliton on a warped product manifold and show that if the base manifold is oriented, compact and of constant scalar curvature, the soliton on the product manifold gives a lower bound for its scalar curvature.

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41

Sakurai, Yohei. "Rigidity of manifolds with boundary under a lower Bakry-Émery Ricci curvature bound." Tohoku Mathematical Journal 71, no. 1 (March 2019): 69–109. http://dx.doi.org/10.2748/tmj/1552100443.

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42

Tadano, Homare. "A lower diameter bound for compact domain manifolds of shrinking Ricci-harmonic solitons." Kodai Mathematical Journal 38, no. 2 (June 2015): 302–9. http://dx.doi.org/10.2996/kmj/1436403892.

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43

Villemonais, Denis. "Lower Bound for the Coarse Ricci Curvature of Continuous-Time Pure-Jump Processes." Journal of Theoretical Probability 33, no. 2 (May 20, 2019): 954–91. http://dx.doi.org/10.1007/s10959-019-00918-9.

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44

Sakurai, Yohei. "Rigidity Phenomena in Manifolds with Boundary Under a Lower Weighted Ricci Curvature Bound." Journal of Geometric Analysis 29, no. 1 (October 24, 2018): 1–32. http://dx.doi.org/10.1007/s12220-017-9871-7.

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45

Li, Chang-Jun, and Xiang Gao. "A new proof of the bound for the first Dirichlet eigenvalue of the Laplacian operator." Analele Universitatii "Ovidius" Constanta - Seria Matematica 22, no. 2 (June 1, 2014): 129–40. http://dx.doi.org/10.2478/auom-2014-0038.

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AbstractIn this paper, we present a new proof of the upper and lower bound estimates for the first Dirichlet eigenvalue $\lambda _1^D \left({B\left({p,r} \right)} \right)$ of Laplacian operator for the manifold with Ricci curvature Rc ≥ −K, by using Li-Yau’s gradient estimate for the heat equation.

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46

Fernández-López, Manuel, and Eduardo García-Río. "A sharp lower bound for the scalar curvature of certain steady gradient Ricci solitons." Proceedings of the American Mathematical Society 141, no. 6 (February 7, 2013): 2145–48. http://dx.doi.org/10.1090/s0002-9939-2013-11675-8.

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47

Kitabeppu, Yu. "Lower bound of coarse Ricci curvature on metric measure spaces and eigenvalues of Laplacian." Geometriae Dedicata 169, no. 1 (March 10, 2013): 99–107. http://dx.doi.org/10.1007/s10711-013-9844-3.

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48

Yang, DaGang. "Lower bound estimates of the first eigenvalue for compact manifolds with positive Ricci curvature." Pacific Journal of Mathematics 190, no. 2 (October 1, 1999): 383–98. http://dx.doi.org/10.2140/pjm.1999.190.383.

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49

Cai, Kairen. "Global pinching theorems of submanifolds in spheres." International Journal of Mathematics and Mathematical Sciences 31, no. 3 (2002): 183–91. http://dx.doi.org/10.1155/s0161171202106247.

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LetMbe a compact embedded submanifold with parallel mean curvature vector and positive Ricci curvature in the unit sphereS n+p(n≥2 ,p≥1). By using the Sobolev inequalities of P. Li (1980) toLpestimate for the square lengthσof the second fundamental form and the norm of a tensorΦ, related to the second fundamental form, we set up some rigidity theorems. Denote by‖σ‖ptheLpnorm ofσandHthe constant mean curvature ofM. It is shown that there is a constantCdepending only onn,H, andkwhere(n−1) kis the lower bound of Ricci curvature such that if‖σ‖ n/2<C, thenMis a totally umbilic hypersurface in the sphereS n+1.

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50

Datar, Ved, Harish Seshadri, and Jian Song. "Metric rigidity of Kähler manifolds with lower Ricci bounds and almost maximal volume." Proceedings of the American Mathematical Society 149, no. 8 (May 18, 2021): 3569–74. http://dx.doi.org/10.1090/proc/15473.

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In this short note we prove that a Kähler manifold with lower Ricci curvature bound and almost maximal volume is Gromov-Hausdorff close to the projective space with the Fubini-Study metric. This is done by combining the recent results on holomorphic rigidity of such Kähler manifolds (see Gang Liu [Asian J. Math. 18 (2014), 69–99]) with the structure theorem of Tian-Wang (see Gang Tian and Bing Wang [J. Amer. Math. Soc 28 (2015), 1169–1209]) for almost Einstein manifolds. This can be regarded as the complex analog of the result on Colding on the shape of Riemannian manifolds with almost maximal volume.

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