Abstract
Let M be a closed surface. For a metric g on M, denote the Laplace-Beltrami operator by Δ = Δ g . We define trace \({\Delta^{-1} = \int_M m(p)dA}\) , where dA is the area element for g and m(p) is the Robin constant at the point \({p \in M}\) , that is the value of the Green function G(p, q) at q = p after the logarithmic singularity has been subtracted off. Since trace Δ−1 can also be obtained by regularization of the spectral zeta function, it is a spectral invariant. Heuristically it represents the sum of squares of the wavelengths of the surface. We define the Δ-mass of (M, g) to equal \({({\rm trace} \Delta_g^{-1} - {\rm trace} \Delta_{S^{2},A}^{-1})/A}\) , where \({\Delta_{S^2,A}}\) is the Laplacian on the round sphere of area A. This is an analog for closed surfaces of the ADM mass from general relativity. We show that if M has positive genus, the minimum of the Δ-mass on each conformal class is negative and attained by a smooth metric. For this minimizing metric, there is a sharp logarithmic Hardy-Littlewood-Sobolev inequality and a Moser-Trudinger-Onofri type inequality.
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References
Beckner W.: Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality. Annals of Math. 138, 213–242 (1993)
Carlen E., Loss M.: Competing symmetries, the logarithmic HLS inequality and Onofri’s inequality on S n. Geom. Funct. Anal. 2, 90–104 (1992)
Chang S.-Y.A.: Conformal invariants and partial differential equations. Bull. Amer. Math. Soc. 42, 365–393 (2005)
Chen C.-C., Lin C.-S.: Sharp estimates for solutions of multi-bubbles in compact Riemann surfaces. Comm. Pure Appl. Math. 55(6), 728–771 (2002)
Ding W., Jost J., Li J., Wang G.: The differential equation Δu = 8π − 8πhe u on a compact Riemann surface. Asian J. Math. 1, 230–248 (1997)
Doyle, P., Steiner, J.: Spectral invariants and playing hide and seek on surfaces. Preprint, available at http://www.cims.nyu.edu/~steiner/hideandseek.pdf, 2005
Doyle, P., Steiner, J.: Blowing bubbles on the torus. Preprint, available at http://www.cims.nyu.edu/~steiner/torus.pdf, 2005
Fay T.: Theta functions on Riemann surfaces. Ann. Math. 119, 387 (1994)
Lin C.-S., Lucia M.: Uniqueness of solutions for a mean field equation on the torus. J. Diff. Eqs. 229(1), 172–185 (2006)
Lin C.-S., Lucia M.: One-dimensional symmetry of periodic minimizers for a mean field equation. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 6(2), 269–290 (2007)
Morpurgo C.: The logarithmic Hardy-Littlewood-Sobolev inequality and extremals of zeta functions on S n. Geom. Funct. Anal. 6, 146–171 (1996)
Morpurgo, C.: Zeta functions on S 2. Extremal Riemann surfaces (San Francisco, 1995), Contemp. Math., 201, Providence, RI: Amer. Math. Soc., 1997, pp. 213–225
Morpurgo C.: Sharp inequalities for functional integrals and traces of conformally invariant operators. Duke Math. J. 114, 477–553 (2002)
Nolasco M., Tarantello G.: On a sharp Sobolev-type inequality on two-dimensional compact manifolds. Arch. Ration. Mech. Anal. 145, 161–195 (1998)
Okikiolu K.: Extremals for Logarithmic HLS inequalities on compact manifolds. GAFA 107(5), 1655–1684 (2008)
Okikiolu K.: A negative mass theorem for the 2-torus. Commun. Math. Phys. 284(3), 775–802 (2008)
Onofri E.: On the positivity of the effective action in a theory of random surfaces. Commun. Math. Phys. 86, 321–326 (1982)
Osgood B., Phillips R., Sarnak P.: Extremals of determinants of laplacians. J. Funct. Anal. 80, 148–211 (1988)
Steiner, J.: Green’s Functions, Spectral Invariants, and a Positive Mass on Spheres. Ph. D. Dissertation, University of California San Diego, June 2003
Steiner J.: A geometrical mass and its extremal properties for metrics on s 2. Duke Math. J. 129, 63–86 (2005)
Wentworth R.: The asymptotics of the Arakelov-Green’s function and Faltings’ delta invariant. Commun. Math. Phys. 137, 427–459 (1991)
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I would like to thank Richard Wentworth for helpful discussions.
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Communicated by S. Zelditch
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Okikiolu, K. A Negative Mass Theorem for Surfaces of Positive Genus. Commun. Math. Phys. 290, 1025–1031 (2009). https://doi.org/10.1007/s00220-008-0722-z
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DOI: https://doi.org/10.1007/s00220-008-0722-z