Eigenvalue estimates for 3-Sasaki structures
-
Paul-Andi Nagy
Abstract
We obtain new lower bounds for the first non-zero eigenvalue of the scalar sub-Laplacian for 3-Sasaki metrics, improving the Lichnerowicz–Obata-type estimates by Ivanov, Petkov and Vassilev (2013, 2014). The limiting eigenspace is fully described in terms of the automorphism algebra. Our results can be thought of as an analogue of the Lichnerowicz–Matsushima estimate for Kähler–Einstein metrics. In dimension 7, if the automorphism algebra is non-vanishing, we also compute the second eigenvalue for the sub-Laplacian and construct explicit eigenfunctions. In addition, for all metrics in the canonical variation of the 3-Sasaki metric we give a lower bound for the spectrum of the Riemannian Laplace operator, depending only on scalar curvature and dimension. We also strengthen a result pertaining to the growth rate of harmonic functions, due to Conlon, Hein and Sun (2013, 2017), in the case of hyperkähler cones. In this setup we also describe the space of holomorphic functions.
Funding statement: This research has been financially supported by the Special Priority Program SPP 2026 “Geometry at Infinity” funded by the DFG.
Acknowledgements
It is a pleasure to thank Craig van Coevering for many useful email exchanges and the reviewer for suggestions on how to improve presentation.
References
[1] D. V. Alekseevsky, S. Marchiafava and M. Pontecorvo, Spectral properties of the twistor fibration of a quaternion Kähler manifold, J. Math. Pures Appl. (9) 79 (2000), no. 1, 95–110. 10.1016/S0021-7824(00)00145-8Search in Google Scholar
[2] F. Baudoin and N. Garofalo, Curvature-dimension inequalities and Ricci lower bounds for sub-Riemannian manifolds with transverse symmetries, J. Eur. Math. Soc. (JEMS) 19 (2017), no. 1, 151–219. 10.4171/JEMS/663Search in Google Scholar
[3] F. Baudoin and B. Kim, The Lichnerowicz–Obata theorem on sub-Riemannian manifolds with transverse symmetries, J. Geom. Anal. 26 (2016), no. 1, 156–170. 10.1007/s12220-014-9542-xSearch in Google Scholar
[4] F. Baudoin, B. Kim and J. Wang, Transverse Weitzenböck formulas and curvature dimension inequalities on Riemannian foliations with totally geodesic leaves, Comm. Anal. Geom. 24 (2016), no. 5, 913–937. 10.4310/CAG.2016.v24.n5.a1Search in Google Scholar
[5] N. Berline, E. Getzler and M. Vergne, Heat kernels and Dirac operators, Grundlehren Text Editions, Springer, Berlin 2004. Search in Google Scholar
[6] A. L. Besse, Einstein manifolds, Class. Math., Springer, Berlin 2008. Search in Google Scholar
[7] C. Boyer and K. Galicki, 3-Sasakian manifolds, Surveys in differential geometry: Essays on Einstein manifolds, Surv. Differ. Geom. 6, International Press, Boston (1999), 123–184. 10.4310/SDG.2001.v6.n1.a6Search in Google Scholar
[8] C. P. Boyer and K. Galicki, Sasakian geometry, Oxford Math. Monogr., Oxford University, Oxford 2008. 10.1093/acprof:oso/9780198564959.001.0001Search in Google Scholar
[9] H.-D. Cao and C. He, Linear stability of Perelman’s ν-entropy on symmetric spaces of compact type, J. reine angew. Math. 709 (2015), 229–246. 10.1515/crelle-2013-0096Search in Google Scholar
[10] J. Cheeger, Spectral geometry of singular Riemannian spaces, J. Differential Geom. 18 (1983), no. 4, 575–657. 10.4310/jdg/1214438175Search in Google Scholar
[11] J. Cheeger and G. Tian, On the cone structure at infinity of Ricci flat manifolds with Euclidean volume growth and quadratic curvature decay, Invent. Math. 118 (1994), no. 3, 493–571. 10.1007/BF01231543Search in Google Scholar
[12] R. J. Conlon and H.-J. Hein, Asymptotically conical Calabi–Yau manifolds, I, Duke Math. J. 162 (2013), no. 15, 2855–2902. 10.1215/00127094-2382452Search in Google Scholar
[13] M. Fernández, A. Fino, L. Ugarte and R. Villacampa, Strong Kähler with torsion structures from almost contact manifolds, Pacific J. Math. 249 (2011), no. 1, 49–75. 10.2140/pjm.2011.249.49Search in Google Scholar
[14] A. Futaki, H. Ono and G. Wang, Transverse Kähler geometry of Sasaki manifolds and toric Sasaki–Einstein manifolds, J. Differential Geom. 83 (2009), no. 3, 585–635. 10.4310/jdg/1264601036Search in Google Scholar
[15] H.-J. Hein and S. Sun, Calabi–Yau manifolds with isolated conical singularities, Publ. Math. Inst. Hautes Études Sci. 126 (2017), 73–130. 10.1007/s10240-017-0092-1Search in Google Scholar
[16] L. Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147–171. 10.1007/BF02392081Search in Google Scholar
[17] S. Ivanov, A. Petkov and D. Vassilev, The sharp lower bound of the first eigenvalue of the sub-Laplacian on a quaternionic contact manifold in dimension seven, Nonlinear Anal. 93 (2013), 51–61. 10.1016/j.na.2013.07.011Search in Google Scholar
[18] S. Ivanov, A. Petkov and D. Vassilev, The sharp lower bound of the first eigenvalue of the sub-Laplacian on a quaternionic contact manifold, J. Geom. Anal. 24 (2014), no. 2, 756–778. 10.1007/s12220-012-9354-9Search in Google Scholar
[19] C. LeBrun, Fano manifolds, contact structures, and quaternionic geometry, Internat. J. Math. 6 (1995), no. 3, 419–437. 10.1142/S0129167X95000146Search in Google Scholar
[20] J. Lee and K. Richardson, Lichnerowicz and Obata theorems for foliations, Pacific J. Math. 206 (2002), no. 2, 339–357. 10.2140/pjm.2002.206.339Search in Google Scholar
[21] S.-Y. Li, D. N. Son and X. Wang, A new characterization of the CR sphere and the sharp eigenvalue estimate for the Kohn Laplacian, Adv. Math. 281 (2015), 1285–1305. 10.1016/j.aim.2015.06.008Search in Google Scholar
[22]
P.-A. Nagy and U. Semmelmann,
The
[23] L. P. Rothschild and E. M. Stein, Hypoelliptic differential operators and nilpotent groups, Acta Math. 137 (1976), no. 3–4, 247–320. 10.1007/BF02392419Search in Google Scholar
[24] U. Semmelmann and G. Weingart, Vanishing theorems for quaternionic Kähler manifolds, J. reine angew. Math. 544 (2002), 111–132. 10.1515/crll.2002.019Search in Google Scholar
[25] U. Semmelmann and G. Weingart, An upper bound for a Hilbert polynomial on quaternionic Kähler manifolds, J. Geom. Anal. 14 (2004), no. 1, 151–170. 10.1007/BF02921870Search in Google Scholar
[26] R. S. Strichartz, Sub-Riemannian geometry, J. Differential Geom. 24 (1986), no. 2, 221–263. 10.4310/jdg/1214440436Search in Google Scholar
[27] C. van Coevering, Private comunication, 27.04.2021. Search in Google Scholar
© 2023 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Closed binomial edge ideals
- Eigenvalue estimates for 3-Sasaki structures
- The Yamabe flow on asymptotically flat manifolds
- Positive scalar curvature on manifolds with fibered singularities
- A Hitchin connection on nonabelian theta functions for parabolic 𝐺-bundles
- The Lorentzian Lichnerowicz conjecture for real-analytic, three-dimensional manifolds
- Regularity for convex viscosity solutions of Lagrangian mean curvature equation
- Capillary surfaces: Stability, index and curvature estimates
- Sign-changing solution for an overdetermined elliptic problem on unbounded domain
Articles in the same Issue
- Frontmatter
- Closed binomial edge ideals
- Eigenvalue estimates for 3-Sasaki structures
- The Yamabe flow on asymptotically flat manifolds
- Positive scalar curvature on manifolds with fibered singularities
- A Hitchin connection on nonabelian theta functions for parabolic 𝐺-bundles
- The Lorentzian Lichnerowicz conjecture for real-analytic, three-dimensional manifolds
- Regularity for convex viscosity solutions of Lagrangian mean curvature equation
- Capillary surfaces: Stability, index and curvature estimates
- Sign-changing solution for an overdetermined elliptic problem on unbounded domain
