×

Spectral properties of reducible conical metrics. (English) Zbl 1470.30034

Summary: We show that the monodromy of a spherical conical metric \(g\) is reducible if and only if the metric \(g\) has a real-valued eigenfunction with eigenvalue 2 for the holomorphic extension \(\Delta_g^{\text{Hol}}\) of the associated Laplace-Beltrami operator. Such an eigenfunction produces a meromorphic vector field, which is then related to the developing maps of the conical metric. We also give a lower bound of the first nonzero eigenvalue of \(\Delta_g^{\text{Hol}}\), together with a complete classification of the dimension of the space of real-valued 2-eigenfunctions for \(\Delta_g^{\text{Hol}}\) depending on the monodromy of the metric \(g\). This paper can be seen as a new connection between the complex analysis method and the PDE approach in the study of spherical conical metrics.

MSC:

30F45 Conformal metrics (hyperbolic, Poincaré, distance functions)
34M35 Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] W. Ballmann, Lectures on Kähler Manifolds, ESI Lect. Math. Phys. European Mathematical Society (EMS), Zürich, 2006. · Zbl 1101.53042 · doi:10.4171/025
[2] D. Bartolucci, F. De Marchis, and A. Malchiodi, Supercritical conformal metrics on surfaces with conical singularities, Int. Math. Res. Not. IMRN 24 (2011), 5625-5643. · Zbl 1254.30066 · doi:10.1093/imrn/rnq285
[3] J. Brüning and R. Seeley, Regular singular asymptotics, Adv. in Math. 58 (1985), no. 2, 133-148. · Zbl 0593.47047 · doi:10.1016/0001-8708(85)90114-8
[4] J. Brüning and R. Seeley, An index theorem for first order regular singular operators, Amer. J. Math. 110 (1988), no. 4, 659-714. · Zbl 0664.58035 · doi:10.2307/2374646
[5] A. Carlotto, On the solvability of singular Liouville equations on compact surfaces of arbitrary genus, Trans. Amer. Math. Soc. 366 (2014), no. 3, 1237-1256. · Zbl 1306.58005 · doi:10.1090/S0002-9947-2013-05847-3
[6] Q. Chen, X. Chen, and Y. Wu, The structure of HCMU metric in a K-surface, Int. Math. Res. Not. 16 (2005), 941-958. · Zbl 1090.53036 · doi:10.1155/IMRN.2005.941
[7] J. Cheeger, On the spectral geometry of spaces with cone-like singularities, Proc. Nat. Acad. Sci. USA 76 (1979), no. 5, 2103-2106. · Zbl 0411.58003 · doi:10.1073/pnas.76.5.2103
[8] X. Chen, Obstruction to the existence of metric whose curvature has umbilical Hessian in a K-surface, Comm. Anal. Geom. 8 (2000), no. 2, 267-299. · Zbl 0971.53029 · doi:10.4310/CAG.2000.v8.n2.a2
[9] Z. Chen, personal communication, 2019.
[10] Z. Chen, T.-J. Kuo, and C.-S. Lin, Existence and non-existence of solutions of the mean field equations on flat tori, Proc. Amer. Math. Soc. 145 (2017), no. 9, 3989-3996. · Zbl 1373.35108 · doi:10.1090/proc/13543
[11] Q. Chen, B. Li, J. Song, and B. Xu, Jenkins-Strebel differentials and reducible cone spherical metrics on compact Riemann surfaces, in preparation.
[12] C.-L. Chai, C.-S. Lin, and C.-L. Wang, Mean field equations, hyperelliptic curves and modular forms: I, Camb. J. Math. 3 (2015), nos. 1-2, 127-274. · Zbl 1327.35116 · doi:10.4310/CJM.2015.v3.n1.a3
[13] Q. Chen and Y. Wu, Character 1-form and the existence of an HCMU metric, Math. Ann. 351 (2011), no. 2, 327-345. · Zbl 1227.53076 · doi:10.1007/s00208-010-0598-z
[14] Q. Chen, W. Wang, Y. Wu, and B. Xu, Conformal metrics with constant curvature one and finitely many conical singularities on compact Riemann surfaces, Pacific J. Math. 273 (2015), no. 1, 75-100. · Zbl 1308.30050 · doi:10.2140/pjm.2015.273.75
[15] Q. Chen, Y. Wu, and B. Xu, On one-dimensional and singular Calabi’s extremal metrics whose Gauss curvatures have nonzero umbilical Hessians, Israel J. Math. 208 (2015), no. 1, 385-412. · Zbl 1327.30046 · doi:10.1007/s11856-015-1204-6
[16] S. Dey, Spherical metrics with conical singularities on 2-spheres, Geom. Dedicata 196 (2018), 53-61. · Zbl 1404.53047 · doi:10.1007/s10711-017-0306-1
[17] A. Eremenko and A. Gabrielov, On metrics of curvature 1 with four conic singularities on tori and on the sphere, Illinois J. Math. 59 (2015), no. 4, 925-947. · Zbl 1366.30029 · doi:10.1215/ijm/1488186015
[18] A. Eremenko, A. Gabrielov, and V. Tarasov, Spherical quadrilaterals with three non-integer angles, Zh. Mat. Fiz. Anal. Geom. 12 (2016), no. 2, 134-167. · Zbl 1364.30048 · doi:10.15407/mag12.02.134
[19] A. Eremenko, Metrics of positive curvature with conic singularities on the sphere, Proc. Amer. Math. Soc. 132 (2004), no. 11, 3349-3355. · Zbl 1053.53025 · doi:10.1090/S0002-9939-04-07439-8
[20] A. Eremenko, Co-axial monodromy, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), XX (2020), no. 2, 619-634. · Zbl 1481.57032 · doi:10.2422/2036-2145.201706_022
[21] A. Eremenko, Metrics of constant positive curvature with four conic singularities on the sphere, Proc. Amer. Math Soc. 148 (2020), no. 9, 3957-3965. · Zbl 1448.34161 · doi:10.1090/proc/15012
[22] Y. Feng, Y. Q. Shi, and B. Xu, On the explicit expression of a conformal metric of constant curvature one near a conical singularity, J. Univ. Sci. Technol. China 47 (2017), no. 6, 455-458. · Zbl 1399.32017 · doi:10.3969/j.issn.0253-2778.2017.06.001
[23] J. B. Gil, T. Krainer, and G. A. Mendoza, Resolvents of elliptic cone operators, J. Funct. Anal. 241 (2006), no. 1, 1-55. · Zbl 1109.58023 · doi:10.1016/j.jfa.2006.07.010
[24] J. B. Gil, T. Krainer, and G. A. Mendoza, Geometry and spectra of closed extensions of elliptic cone operators, Canad. J. Math. 59 (2007), no. 4, 742-794. · Zbl 1126.58014 · doi:10.4153/CJM-2007-033-7
[25] J. B. Gil and G. A. Mendoza, Adjoints of elliptic cone operators, Amer. J. Math. 125 (2003), no. 2, 357-408. · Zbl 1030.58012 · doi:10.1353/ajm.2003.0012
[26] J. Hersch, Quatre propriétés isopérimétriques de membranes sphériques homogènes, C. R. Acad. Sci. Paris Sér. A-B 270 (1970), A1645-A1648. · Zbl 0224.73083
[27] L. Hillairet, “Spectral theory of translation surfaces: A short introduction” in Actes du Séminaire de Théorie Spectrale et Géometrie, Année 2009-2010, Sémin. Théor. Spectr. Géom. 28, Univ. Grenoble I, Saint-Martin-d’Hères, 2010, 51-62. · Zbl 1404.58050 · doi:10.5802/tsg.278
[28] L. Hillairet and A. Kokotov, Isospectrality, comparison formulas for determinants of Laplacian and flat metrics with non-trivial holonomy, Proc. Amer. Math. Soc. 145 (2017), no. 9, 3915-3928. · Zbl 1380.30031 · doi:10.1090/proc/13494
[29] M. Kapovich, Branched covers between spheres and polygonal inequalities in simplicial trees, https://math.ucdavis.edu/ kapovich/EPR/covers.pdf, 2017.
[30] M. Karpukhin, N. Nadirashvili, A. V. Penskoi, and I. Polterovich, An isoperimetric inequality for Laplace eigenvalues on the sphere, accepted J. Differential Geom. · Zbl 1471.53056
[31] M. Lesch, Operators of Fuchs type, conical singularities, and asymptotic methods, Teubner-Texte zur Mathematik [Teubner Texts in Mathematics] 136, B. G. Teubner Verlagsgesellschaft, Stuttgart, 1997. · Zbl 1156.58302
[32] A. Lichnerowicz, Géométrie des groupes de transformations, Travaux et Recherches Mathématiques III. Dunod, Paris, 1958. · Zbl 0096.16001
[33] F. Luo and G. Tian, Liouville equation and spherical convex polytopes, Proc. Amer. Math. Soc. 116 (1992), no. 4, 1119-1129. · Zbl 0806.53012 · doi:10.2307/2159498
[34] C.-S. Lin and C.-L. Wang, Elliptic functions, Green functions and the mean field equations on tori, Ann. of Math. (2), 172 (2010), no. 2, 911-954. · Zbl 1207.35011 · doi:10.4007/annals.2010.172.911
[35] C.-S. Lin and X. Zhu, Explicit construction of extremal Hermitian metrics with finite conical singularities on \[{S^2} \], Comm. Anal. Geom. 10 (2002), no. 1, 177-216. · Zbl 1021.58008 · doi:10.4310/CAG.2002.v10.n1.a8
[36] Y. Matsushima, Sur la structure du groupe d’homéomorphismes analytiques d’une certaine variété kaehlerienne, Nagoya Math. J. 11 (1957), 145-150. · Zbl 0091.34803 · doi:10.1017/S0027763000002026
[37] R. C. McOwen, Point singularities and conformal metrics on Riemann surfaces, Proc. Amer. Math. Soc. 103 (1988), no. 1, 222-224. · Zbl 0657.30033 · doi:10.2307/2047555
[38] R. B. Melrose, The Atiyah-Patodi-Singer index theorem, Res. Notes Math. 4, A. K. Peters, Wellesley, MA, 1993. · Zbl 0796.58050 · doi:10.1016/0377-0257(93)80040-i
[39] E. A. Mooers, Heat kernel asymptotics on manifolds with conic singularities, J. Anal. Math. 78 (1999), 1-36. · Zbl 0981.58022 · doi:10.1007/BF02791127
[40] G. Mondello and D. Panov, Spherical metrics with conical singularities on a 2-sphere: Angle constraints, Int. Math. Res. Not. IMRN 16 (2016), 4937-4995. · Zbl 1446.53027 · doi:10.1093/imrn/rnv300
[41] G. Mondello and D. Panov, Spherical surfaces with conical points: Systole inequality and moduli spaces with many connected components, Geom. Funct. Anal. 29 (2019), no. 4, 1110-1193. · Zbl 1447.58013 · doi:10.1007/s00039-019-00506-3
[42] R. Mazzeo and H. Weiss, “Teichmüller theory for conic surfaces” in Geometry, Analysis and Probability, Progr. Math. 310, Birkhäuser/Springer, Cham, 2017. · Zbl 1444.32015
[43] R. Mazzeo and X. Zhu, Conical metrics on Riemann surfaces, I: The compactified configuration space and regularity, Geom. Topol. 24 (2020), no. 1, 309-372. · Zbl 1443.53018 · doi:10.2140/gt.2020.24.309
[44] R. Mazzeo and X. Zhu, Conical metrics on Riemann surfaces, II: Spherical metrics, accepted Int. Math. Res. Not.
[45] M. Obata, Certain conditions for a Riemannian manifold to be isometric with a sphere, J. Math. Soc. Japan 14, 333-340. · Zbl 0115.39302 · doi:10.2969/jmsj/01430333
[46] M. Reed and B. Simon, Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-Adjointness, Academic Press, New York, 1975. · Zbl 0308.47002
[47] M. Reed and B. Simon. Methods of Modern Mathematical Physics. I, 2nd ed. Academic Press, New York, 1980. Functional analysis. · Zbl 0459.46001
[48] J. Song, Y. Cheng, B. Li, and B. Xu, Drawing cone spherical metrics via Strebel differentials, Int. Math. Res. Not. IMRN 11 (2020), 3341-3363. · Zbl 1487.53056 · doi:10.1093/imrn/rny103
[49] J. Song and B. Xu, On rational functions with more than three branch points, Algebra Colloq. 27 (2020), no. 2, 231-246. · Zbl 1481.20009 · doi:10.1142/S100538672000019X
[50] M. Troyanov, Prescribing curvature on compact surfaces with conical singularities, Trans. Amer. Math. Soc. 324 (1991), no. 2, 793-821. · Zbl 0724.53023 · doi:10.2307/2001742
[51] M. Umehara and K. Yamada, Metrics of constant curvature 1 with three conical singularities on the 2-sphere, Illinois J. Math. 44 (2000), no. 1, 72-94. · Zbl 0958.30029 · doi:10.1215/ijm/1255984594
[52] X. Zhu, Rigidity of a family of spherical conical metrics, New York J. Math. 26 (2020), 272-284. · Zbl 1435.51004
[53] X. Zhu, Spherical conic metrics and realizability of branched covers, Proc. Amer. Math. Soc. 147 (2019), no. 4, 1805-1815 · Zbl 1420.57008 · doi:10.1090/proc/14318
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.