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Advances in Geometry

Managing Editor: Grundhöfer, Theo / Joswig, Michael

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Eigenvalues of the weighted Laplacian under the extended Ricci flow

Abimbola Abolarinwa
Published Online: 2018-10-25 | DOI: https://doi.org/10.1515/advgeom-2018-0022

Abstract

Let φ = − ∇φ∇ be a symmetric diffusion operator with an invariant weighted volume measure = eφ on an n-dimensional compact Riemannian manifold (M, g), where g = g(t) solves the extended Ricci flow. We study the evolution and monotonicity of the first nonzero eigenvalue of φ and we obtain several monotone quantities along the extended Ricci flow and its volume preserving version under some technical assumption. We also show that the eigenvalues diverge in a finite time for n ≥ 3. Our results are natural extensions of some known results for Laplace–Beltrami operators under various geometric flows.

Keywords: Witten–Laplacian; eigenvalues; Ricci flow; monotonicity; curvature

MSC 2010: Primary 53C21; 53C44; Secondary 35P30; 58J35

References

  • [1]

    A. Abolarinwa, Evolution and monotonicity of the first eigenvalue of p-Laplacian under the Ricci-harmonic flow. J. Appl. Anal. 21 (2015), 147–160. MR3430912 Zbl 1329.53052Google Scholar

  • [2]

    A. Abolarinwa, J. Mao, The first eigenvalue of the p-Laplacian on time dependent Riemannian metrics. Preprint 2016, arXiv:1605.01882 [math.DG]Google Scholar

  • [3]

    D. Bakry, M. Émery, Diffusions hypercontractives. In: Séminaire de probabilités, XIX, 1983/84, volume 1123 of Lecture Notes in Math., 177–206, Springer 1985. MR889476 Zbl 0561.60080Google Scholar

  • [4]

    M. Băileşteanu, H. Tran, Heat kernel estimates under the Ricci-harmonic map flow. Proc. Edinb. Math. Soc. (2) 60 (2017), 831–857. MR3715688Google Scholar

  • [5]

    X. Cao, Eigenvalues of (Δ+R2) on manifolds with nonnegative curvature operator. Math. Ann. 337 (2007), 435–441. MR2262792 Zbl 1105.53051Google Scholar

  • [6]

    X. Cao, First eigenvalues of geometric operators under the Ricci flow. Proc. Amer. Math. Soc. 136 (2008), 4075–4078. MR2425749 Zbl 1166.58007Google Scholar

  • [7]

    X. Cao, S. Hou, J. Ling, Estimate and monotonicity of the first eigenvalue under the Ricci flow. Math. Ann. 354 (2012), 451–463. MR2965250 Zbl 1252.53075Google Scholar

  • [8]

    X. Cheng, D. Zhou, Eigenvalues of the drifted Laplacian on complete metric measure spaces. Commun. Contemp. Math. 19 (2017), 1650001, 17. MR3575913 Zbl 1360.58022Google Scholar

  • [9]

    B. Chow, D. Knopf, The Ricci flow: an introduction, volume 110 of Mathematical Surveys and Monographs. Amer. Math. Soc. 2004. MR2061425 Zbl 1086.53085Google Scholar

  • [10]

    S. Fang, H. Xu, P. Zhu, Evolution and monotonicity of eigenvalues under the Ricci flow. Sci. China Math. 58 (2015), 1737–1744. MR3368179 Zbl 1327.53084Google Scholar

  • [11]

    A. Futaki, H. Li, X.-D. Li, On the first eigenvalue of the Witten-Laplacian and the diameter of compact shrinking solitons. Ann. Global Anal. Geom. 44 (2013), 105–114. MR3073582 Zbl 1273.58018Google Scholar

  • [12]

    A. Grigor’yan, Heat kernel and analysis on manifolds, volume 47 of AMS/IP Studies in Advanced Mathematics. Amer. Math. Soc. 2009. MR2569498 Zbl 1206.58008Google Scholar

  • [13]

    H. Guo, R. Philipowski, A. Thalmaier, Entropy and lowest eigenvalue on evolving manifolds. Pacific J. Math. 264 (2013), 61–81. MR3079761 Zbl 1275.53058Google Scholar

  • [14]

    H. X. Guo, R. Philipowski, A. Thalmaier, On gradient solitons of the Ricci-harmonic flow. Acta Math. Sin. (Engl. Ser.) 31 (2015), 1798–1804. MR3406677 Zbl 1330.53085Google Scholar

  • [15]

    R. S. Hamilton, Three-manifolds with positive Ricci curvature. J. Differential Geom. 17 (1982), 255–306. MR664497 Zbl 0504.53034Google Scholar

  • [16]

    G. Huang, Z. Li, Monotonicity formulas of eigenvalues and energy functionals along the rescaled List’s extended Ricci flow. Preprint 2015, arXiv:1511.08529v1 [math.DG]Google Scholar

  • [17]

    J.-F. Li, Eigenvalues and energy functionals with monotonicity formulae under Ricci flow. Math. Ann. 338 (2007), 927–946. MR2317755 Zbl 1127.53059Google Scholar

  • [18]

    J.-F. Li, Monotonicity formulae under rescaled Ricci flow. Preprint 2007, arXiv:math/07010.5328 [math.DG]Google Scholar

  • [19]

    Y. Li, Eigenvalues and entropies under the harmonic-Ricci flow. Pacific J. Math. 267 (2014), 141–184. MR3163480 Zbl 1312.53088Google Scholar

  • [20]

    Y. Li, Long time existence and bounded scalar curvature in Ricci-harmonic flow. Preprint 2015, arXiv 1510.05788v2 [math.DG]Google Scholar

  • [21]

    J. Ling, A class of monotone quantities along the Ricci flow. Preprint 2007, arXiv:0710.4291Google Scholar

  • [22]

    B. List, Evolution of an extended Ricci flow system. Comm. Anal. Geom. 16 (2008), 1007–1048. MR2471366 Zbl 1166.53044Google Scholar

  • [23]

    L. Ma, Eigenvalue monotonicity for the Ricci–Hamilton flow. Ann. Global Anal. Geom. 29 (2006), 287–292. MR2248073 Zbl 1099.53046Google Scholar

  • [24]

    L. Ma, Eigenvalue estimates and L1 energy on closed manifolds. Acta Math. Sin. (Engl. Ser.) 30 (2014), 1729–1734. MR3255784 Zbl 1304.58017Google Scholar

  • [25]

    R. Müller, Ricci flow coupled with harmonic map flow. Ann. Sci. Éc. Norm. Supér. (4) 45 (2012), 101–142. MR2961788 Zbl 1247.53082Google Scholar

  • [26]

    T. Oliynyk, V. Suneeta, E. Woolgar, A gradient flow for worldsheet nonlinear sigma models. Nuclear Phys. B 739 (2006), 441–458. MR2214659 Zbl 1109.81058Google Scholar

  • [27]

    G. Perelman, The entropy formula for the Ricci flow and its geometric application. Preprint 2002, arXiv:0211159v1 [math.DG]Google Scholar

  • [28]

    H. Tadano, Gap theorems for Ricci-harmonic solitons. Ann. Global Anal. Geom. 49 (2016), 165–175. MR3464218 Zbl 1335.53090Google Scholar

  • [29]

    H. Tran, Harnack estimates for Ricci flow on a warped product. J. Geom. Anal. 26 (2016), 1838–1862. MR3511460 Zbl 1343.53068Google Scholar

  • [30]

    M. B. Williams, Results on coupled Ricci and harmonic map flows. Adv. Geom. 15 (2015), 7–26. MR3300708 Zbl 1310.53043Google Scholar

  • [31]

    L. Zhao, Eigenvalues of the Laplacian operator under mean curvature flow. Chinese Ann. Math. Ser. A 30 (2009), 539–544. MR2582093 Zbl 1212.53095Google Scholar

  • [32]

    L. Zhao, The first eigenvalue of p-Laplace operator under powers of the mth mean curvature flow. Results Math. 63 (2013), 937–948. MR3057347 Zbl 1270.53089Google Scholar

About the article


Received: 2016-07-22

Revised: 2017-03-24

Published Online: 2018-10-25


Citation Information: Advances in Geometry, ISSN (Online) 1615-7168, ISSN (Print) 1615-715X, DOI: https://doi.org/10.1515/advgeom-2018-0022.

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