We define the "sum of squares of the wavelengths" of a Riemannian surface
(M,g) to be the regularized trace of the inverse of the Laplacian. We normalize
by scaling and adding a constant, to obtain a "mass", which is scale invariant
and vanishes at the round sphere. This is an anlaog for closed surfaces of the
ADM mass from general relativity. We show that if M has positive genus then on
each conformal class, the mass attains a negative minimum. For the minimizing
metric, there is a sharp logarithmic Hardy-Littlewood-Sobolev inequality and a
Moser-Trudinger-Onofri type inequality.Comment: 8 page

B. Osgood

C. Morpurgo

C.-C. Chen

C.-S. Lin

E. Carlen

E. Onofri

J. Steiner

K. Okikiolu

M. Nolasco

R. Wentworth

S.-Y.A. Chang

T. Fay

W. Beckner

W. Ding

English

arXiv

Springer - Publisher Connector

A Negative Mass Theorem for Surfaces of Positive Genus

Digital Object Identifier (DOI) 10.1007/s00220-008-0722-z
Commun. Math. Phys. 290, 1025–1031 (2009) Communications in
Mathematical
Physics
A Negative Mass Theorem for Surfaces of Positive Genus
K. Okikiolu
Department of Mathematics, University of California, San Diego,
9500 Gilman Drive, La Jolla, CA 92093, USA. E-mail: okikiolu@math.ucsd.edu
Received: 5 October 2008 / Accepted: 13 October 2008
Published online: 22 January 2009 – © The Author(s) 2009. This article is published with open access at
Springerlink.com
Abstract: Let M be a closed surface. For a metric g on M , denote the Laplace-Beltrami
operator by  = g . We define trace −1 =
∫
M m(p) d A, where d A is the area ele-
ment for g and m(p) is the Robin constant at the point p ∈ 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 −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 -mass of (M, g) to equal (trace −1g − trace −1S2,A)/A,
where S2,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 -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.
Section 1. Introduction
Let M be a closed Riemann surface and let g be a metric on M compatible with the
complex structure. With respect to complex coordinates z, the metric g is the real part
of the Kähler metric
eu dz ⊗ dz¯,
for some smooth real valued function u on M . The area element is
d A = ie
u
2
dz ∧ dz¯.
Denote the total area by A. The Laplace Beltrami operator for the metric g is given by
 = g = −4e−u∂z∂z¯ .
 The author would like to acknowledge the support of the Institute for Advanced Study.
1026 K. Okikiolu
(This is sometimes called the geometer’s Laplacian: note the sign.) The Green’s function
for the metric g is the smooth real valued function G on M × M\{(p, p) : p ∈ M} such
that
∫
M
G(p, q)  f (q) d A(q) = f (p) − 1
A
∫
M
f d A
for smooth functions f on M . It follows that G is symmetric, ∫ G(p, q) d A(q) = 0,
and
pG(p, q) = − 1A when p = q. (1.1)
For the smooth function f on M we define
−1 f (p) =
∫
M
G(p, q) f (q) d A(q).
If d(p, q) is the geodesic distance between p and q in the metric g then there exists a
smooth function m on M such that
G(p, q) = 1
2π
log d(p, q) + m(p) + O(d(p, q)), as d(p, q) → 0.
The value m(p) is known as the Robin constant at p. We define
trace −1g =
∫
M
mg d A.
This is a spectral invariant for , since it can be obtained from the spectral zeta function
associated to , see [S1,S2,M3], or [Ok1]. Heuristically it represents the sum of squares
of the wavelengths of the surface (up to a constant). It is convenient to normalize to get
a scale invariant quantity. Indeed, define the -mass to be
M(g) =
trace −1g − trace −1S2,A
A
,
where S2,A is the Laplacian for the round metric on S2 with area A. Then the above
results show that M(g) is always positive when g is a metric on S2. In this paper we
show the following:
Theorem 1 (Negative mass theorem for positive genus surfaces). Given a metric g on
the closed surface M of positive genus, there exists a conformal metric eφg such that
M(eφg) < 0. In fact eφg can be chosen to minimize M within the conformal class.
When M is a torus, this was proved in [Ok2]. We remark that Theorem 1 fails on the
sphere. This follows from the logarithmic Hardy-Littlewood-Sobolev inequality for the
sphere [On,CL,B]:
Theorem (Morpurgo [M2]) For a metric g on the sphere S2, the value M(g) is strictly
positive unless g is round.
From [Ok1], we immediately obtain the following corollary to Theorem 1
Negative Mass Theorem for Surfaces of Positive Genus 1027
Corollary 2 (Analogs of Logarithmic HLS inequality and the Moser-Trudinger-Onofri
Inequality for general surfaces). If the metric g minimizes M within its conformal class,
then
1
4π
∫
M
ψ eψ d A − 1
A
∫
M
eψ−1eψ d A ≥ 0
for all functions ψ : M → R with ∫M eψ d A = A such that
∫
M ψ e
ψ d A is finite. Here,
d A and  are associated to g. Moreover, for ψ ∈ C∞(M),
1
16π
∫
M
ψψ d A − log
(
1
A
∫
M
eψ d A
)
+
1
A
∫
M
ψ d A ≥ 0.
For some related results, see [Ch,CheC,DJLW,LL1,LL2,M2,M3,NT,Ok1,Ok2,OPS,
S2].
Remark. The quantity M(g) can be viewed as an analog of the ADM mass. Indeed,
writing K (p) for the Gaussian curvature of g at p, it is shown in [S1,S2], that for any
metric g on the 2-sphere, the natural analog of the ADM mass for metrics on the sphere
is the constant
mg(p) − 12π 
−1 K (p) = 1
A
trace −1g . (1.2)
Although the left-hand side of (1.2) is not constant in general for surfaces of higher
genus, the right-hand side can be thought of as the analog of the mass. (The left-hand
side of (1.2) is constant for the canonical metric, a fact we use in next section.) For a
probabilistic interpretation of trace −1, see [DS1]. There it is shown that trace −1 is
the constant term in an asymptotic expansion in ε, of the time it takes a Brownian particle
starting at a randomly chosen point on the surface to get ε-close to another randomly
chosen point.
Section 2. The Proof
There are two main ingredients in the proof of this result. The first is an identity con-
cerning the Arakelov Green’s function which is used in the construction of the Arakelov
metric. The second is a delicate result on the mean field equation on surfaces proved in
[DJLW]. That paper gives conditions under which a general mean field equation has a
solution. Here we show that for the particular case of the canonical metric on M and the
mean field equation arising from trace −1, the conditions of the [DJLW] theorem are
satisfied. We start by recalling the way that Robin’s constant and the sum of squares of
the wavelengths change under a conformal change of the metric.
Proposition 2.1. Conformal change of the Robin constant. If φ is a smooth function
on M then
meφg(p) = mg(p) +
φ
4π
− 2
Aφ
(−1g eφ)(p) +
1
A2φ
∫
M
eφ−1g eφ d A,
where
Aφ =
∫
M
eφ d A.
For the proof, see for example [S1,S2,M3] or [Ok1].
1028 K. Okikiolu
Proposition 2.2. Conformal change of trace −1 (Morpurgo’s Formula) Ifφ is a smooth
function on M, then
trace −1
eφg =
∫
M
mge
φ d A +
1
4π
∫
M
φ eφ d A − 1
Aφ
∫
M
eφ−1g eφ d A. (2.1)
Now we discuss the Canonical metric and the Arakelov Green’s function. We refer
the reader to [W] and [F] for more details on this subject. If M is a Riemann surface of
genus H , there exists a metric g on M known as the canonical metric which is com-
patible with the complex structure. It is defined by taking the Jacobian embedding of
the Riemann surface M into a 2H -dimensional torus, and pulling back the flat metric
on the torus to M . Indeed, let {A j , B j } be a symplectic homology basis for H1(M, Z)
satisfying the intersection pairings
#[Ai , A j ] = 0, #[Bi , B j ] = 0, #[Ai , B j ] = δi j .
Take a basis θ j for the space of homomorphic 1-forms satisfying
∫
A j
θk = δ jk .
Then the period matrix 
i j given by

i j =
∫
Bi
θ j
is positive definite. The Jacobian variety associated to M is
J (M) = CH/(Zh + 
ZH ).
The Abel map gives an embedding of M into J (M),
I : z →
∫ z
z0
(θ1, . . . , θH ).
The canonical Kähler metric on M is given in terms of local holomorphic coordinates z
by
µ(z) dz ⊗ dz¯, where µ(z) = 1
H
⎛
⎝
H∑
j,k=1
(Im 
)−1jk
dθ j
dz
d θ¯k
d z¯
⎞
⎠ .
The real part of the Kähler metric is the Riemannian metric g. It can be checked that
this metric has unit area. The Green’s function for this metric is known as the Arakelov
Green’s function and the following result is well known.
Proposition 2.3. If M is a closed Riemann surface of genus H and if g is the canonical
metric on M with unit area, then the Robin constant m(p) for g satisfies
m(p) = 2H − 2 + K (p)
2π
.
Negative Mass Theorem for Surfaces of Positive Genus 1029
Proof of Proposition 2.3. The Gaussian curvature K is given by
2∂z∂z¯ log µ = −µK .
The Arakelov Green’s function is given by
G(z, w) = − 1
2π
log |E(z, w)| + 1
2
H∑
j,k=1
(Im 
)−1jk Im(Z − W ) j Im(Z − W )k
+
m(z)
2
− log µ(z)
8π
+
m(w)
2
− log µ(w)
8π
.
Here, Z = I (z) and W = I (w) and E(z, w) is the prime form which plays the role
of z − w, is holomorphic in z and w, and transforms as a (−1/2,−1/2) form in each
variable. We notice that log |E(z, w)| is harmonic, and so from (1.1) we have that for
w = z,
µ(z) = 4∂z∂z¯G(z, w) = Hµ(z) + 2∂z∂z¯m(z) + µ(z)K (z)4π .
Hence we see that
4µ−1∂z∂z¯m = 2 − 2H − K2π .
unionsq
Theorem 1 is now an application of the following result on the mean field equation
which is obtained from Theorem 1.2 of [DJLW] and its proof.
Theorem 2.4. [DJLW] Let (M, g) be a closed surface of unit area and let h be a smooth
positive function on M. Suppose p0 is a point at which 8πm+2 log h attains its maximum
value, and suppose in addition that
 log h(p0) < 8π − 2K (p0).
Then the minimum of the functional
J (u) = 1
16π
∫
M
|∇u|2 d A +
∫
M
u d A − log
∫
M
heu d A (2.2)
over functions u in the Sobolev space H1(M) is attained at a smooth function u satisfying
u = 8πheu − 8π. (2.3)
Moreover, for this minimum point u we have
J (u) < −
(
1 + log π + max
p∈M(4πmg(p) + log h(p))
)
. (2.4)
Proof of Theorem 1. We take g to be the canonical metric on M , and we set
h = e−4πmg .
1030 K. Okikiolu
Then by Proposition 2.3, we have
 log h = −4πmg = 8π − 8π H − 2K < 8π − 2K .
Hence we obtain the conclusion of Theorem 2.4 From (2.3), the function u satisfies
∫
M
heu d A = 1.
From (2.2) and (2.3) we see that
J (u) = 1
2
∫
M
u(heu + 1) d A.
However, writing
φ = u − 4πmg,
we have
∫
M
eφ d A = 1,
and
−1g eφ =
1
8π
(
u −
∫
M
u
)
. (2.5)
Hence from (2.1),
trace −1
eφg =
1
8π
∫
M
u(heu + 1) = J (u)
4π
.
Now from (2.4) and the fact that
trace −1S2,1 =
−1 − log π
4π
,
we see that
trace −1
eφg < trace 
−1
S2,1.
In fact we remark that eφg minimizes trace −1 among unit area metrics in the conformal
class of g. Indeed, from [Ok] Theorem 1, we conclude that the minimum of trace −1
eψ g
among conformal factors with
∫
eψ = 1, must in fact be attained at a metric eψ g with∫
M e
ψ d A = 1, which must also satisfy the Euler-Lagrange equation, namely that the
Robin constant meψ g(p) is constant:
−1g eψ =
1
8π
(
ψ + 4πmg −
∫
M
(ψ + 4πmg) d A
)
.
However, setting v = ψ + 4πmg , we find that v is a critical metric for J , and hence
trace −1
eψ g =
J (v)
4π
≥ J (u)
4π
= trace −1
eφg.
We also remark that in general the metric eφg need not coincide with the canonical
metric or the constant curvature metric or the Arakelov metric, see [Ok2]. On a long
thin rectangular torus, the minimizer is close to being a round sphere with a short worm
hole joining the poles.
Negative Mass Theorem for Surfaces of Positive Genus 1031
Acknowledgement. I would like to thank Richard Wentworth for helpful discussions.
Open Access This article is distributed under the terms of the Creative Commons Attribution Noncommercial
License which permits any noncommercial use, distribution, and reproduction in any medium, provided the
original author(s) and source are credited.
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Communicated by S. Zelditch


Crossref

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

Okikiolu, K.

eScholarship - University of California

UC San DiegoUC San Diego Previously Published WorksTitleA Negative Mass Theorem for Surfaces of Positive GenusPermalinkhttps://escholarship.org/uc/item/35q77069JournalCommunications in Mathematical Physics, 290(3)ISSN1432-0916AuthorOkikiolu, K.Publication Date2009-09-01DOI10.1007/s00220-008-0722-z Peer reviewedeScholarship.org Powered by the California Digital LibraryUniversity of CaliforniaDigital Object Identifier (DOI) 10.1007/s00220-008-0722-zCommun. Math. Phys. 290, 1025–1031 (2009) Communications inMathematicalPhysicsA Negative Mass Theorem for Surfaces of Positive GenusK. OkikioluDepartment of Mathematics, University of California, San Diego,9500 Gilman Drive, La Jolla, CA 92093, USA. E-mail: okikiolu@math.ucsd.eduReceived: 5 October 2008 / Accepted: 13 October 2008Published online: 22 January 2009 – © The Author(s) 2009. This article is published with open access atSpringerlink.comAbstract: Let M be a closed surface. For a metric g on M , denote the Laplace-Beltramioperator by  = g . We define trace−1 =∫M m(p) d A, where d A is the area ele-ment for g and m(p) is the Robin constant at the point p ∈ M , that is the value of theGreen function G(p, q) at q = p after the logarithmic singularity has been subtractedoff. Since trace−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 wavelengthsof the surface. We define the -mass of (M, g) to equal (trace−1g − trace−1S2,A)/A,whereS2,A is the Laplacian on the round sphere of area A. This is an analog for closedsurfaces of the ADM mass from general relativity. We show that if M has positive genus,the minimum of the-mass on each conformal class is negative and attained by a smoothmetric. For this minimizing metric, there is a sharp logarithmic Hardy-Littlewood-Sobolev inequality and a Moser-Trudinger-Onofri type inequality.Section 1. IntroductionLet M be a closed Riemann surface and let g be a metric on M compatible with thecomplex structure. With respect to complex coordinates z, the metric g is the real partof the Kähler metriceu dz ⊗ dz̄,for some smooth real valued function u on M . The area element isd A = ieu2dz ∧ dz̄.Denote the total area by A. The Laplace Beltrami operator for the metric g is given by = g = −4e−u∂z∂z̄ . The author would like to acknowledge the support of the Institute for Advanced Study.1026 K. Okikiolu(This is sometimes called the geometer’s Laplacian: note the sign.) The Green’s functionfor the metric g is the smooth real valued function G on M × M\{(p, p) : p ∈ M} suchthat∫MG(p, q)  f (q) d A(q) = f (p) − 1A∫Mf d Afor smooth functions f on M . It follows that G is symmetric,∫G(p, q) d A(q) = 0,andpG(p, q) = − 1Awhen p = q. (1.1)For the smooth function f on M we define−1 f (p) =∫MG(p, q) f (q) d A(q).If d(p, q) is the geodesic distance between p and q in the metric g then there exists asmooth function m on M such thatG(p, q) = 12πlog d(p, q) + m(p) + O(d(p, q)), as d(p, q) → 0.The value m(p) is known as the Robin constant at p. We definetrace−1g =∫Mmg d A.This is a spectral invariant for, since it can be obtained from the spectral zeta functionassociated to, see [S1,S2,M3], or [Ok1]. Heuristically it represents the sum of squaresof the wavelengths of the surface (up to a constant). It is convenient to normalize to geta scale invariant quantity. Indeed, define the -mass to beM(g) =trace−1g − trace−1S2,AA,where S2,A is the Laplacian for the round metric on S2 with area A. Then the aboveresults show that M(g) is always positive when g is a metric on S2. In this paper weshow the following:Theorem 1 (Negative mass theorem for positive genus surfaces). Given a metric g onthe closed surface M of positive genus, there exists a conformal metric eφg such thatM(eφg) < 0. In fact eφg can be chosen to minimize M within the conformal class.When M is a torus, this was proved in [Ok2]. We remark that Theorem 1 fails on thesphere. This follows from the logarithmic Hardy-Littlewood-Sobolev inequality for thesphere [On,CL,B]:Theorem (Morpurgo [M2]) For a metric g on the sphere S2, the value M(g) is strictlypositive unless g is round.From [Ok1], we immediately obtain the following corollary to Theorem 1Negative Mass Theorem for Surfaces of Positive Genus 1027Corollary 2 (Analogs of Logarithmic HLS inequality and the Moser-Trudinger-OnofriInequality for general surfaces). If the metric g minimizes M within its conformal class,then14π∫Mψ eψ d A − 1A∫Meψ−1eψ d A ≥ 0for all functions ψ : M → R with ∫M eψ d A = A such that∫M ψ eψ d A is finite. Here,d A and  are associated to g. Moreover, for ψ ∈ C∞(M),116π∫Mψψ d A − log(1A∫Meψ d A)+1A∫Mψ d A ≥ 0.For some related results, see [Ch,CheC,DJLW,LL1,LL2,M2,M3,NT,Ok1,Ok2,OPS,S2].Remark. The quantity M(g) can be viewed as an analog of the ADM mass. Indeed,writing K (p) for the Gaussian curvature of g at p, it is shown in [S1,S2], that for anymetric g on the 2-sphere, the natural analog of the ADM mass for metrics on the sphereis the constantmg(p) − 12π−1 K (p) = 1Atrace−1g . (1.2)Although the left-hand side of (1.2) is not constant in general for surfaces of highergenus, the right-hand side can be thought of as the analog of the mass. (The left-handside of (1.2) is constant for the canonical metric, a fact we use in next section.) For aprobabilistic interpretation of trace−1, see [DS1]. There it is shown that trace−1 isthe constant term in an asymptotic expansion in ε, of the time it takes a Brownian particlestarting at a randomly chosen point on the surface to get ε-close to another randomlychosen point.Section 2. The ProofThere are two main ingredients in the proof of this result. The first is an identity con-cerning the Arakelov Green’s function which is used in the construction of the Arakelovmetric. The second is a delicate result on the mean field equation on surfaces proved in[DJLW]. That paper gives conditions under which a general mean field equation has asolution. Here we show that for the particular case of the canonical metric on M and themean field equation arising from trace−1, the conditions of the [DJLW] theorem aresatisfied. We start by recalling the way that Robin’s constant and the sum of squares ofthe wavelengths change under a conformal change of the metric.Proposition 2.1. Conformal change of the Robin constant. If φ is a smooth functionon M thenmeφg(p) = mg(p) +φ4π− 2Aφ(−1g eφ)(p) +1A2φ∫Meφ−1g eφ d A,whereAφ =∫Meφ d A.For the proof, see for example [S1,S2,M3] or [Ok1].1028 K. OkikioluProposition 2.2. Conformal change of trace−1 (Morpurgo’s Formula) Ifφ is a smoothfunction on M, thentrace−1eφg=∫Mmgeφ d A +14π∫Mφ eφ d A − 1Aφ∫Meφ−1g eφ d A. (2.1)Now we discuss the Canonical metric and the Arakelov Green’s function. We referthe reader to [W] and [F] for more details on this subject. If M is a Riemann surface ofgenus H , there exists a metric g on M known as the canonical metric which is com-patible with the complex structure. It is defined by taking the Jacobian embedding ofthe Riemann surface M into a 2H -dimensional torus, and pulling back the flat metricon the torus to M . Indeed, let {A j , B j } be a symplectic homology basis for H1(M,Z)satisfying the intersection pairings#[Ai , A j ] = 0, #[Bi , B j ] = 0, #[Ai , B j ] = δi j .Take a basis θ j for the space of homomorphic 1-forms satisfying∫A jθk = δ jk .Then the period matrix i j given byi j =∫Biθ jis positive definite. The Jacobian variety associated to M isJ (M) = CH/(Zh +ZH ).The Abel map gives an embedding of M into J (M),I : z →∫ zz0(θ1, . . . , θH ).The canonical Kähler metric on M is given in terms of local holomorphic coordinates zbyµ(z) dz ⊗ dz̄, where µ(z) = 1H⎛⎝H∑j,k=1(Im)−1jkdθ jdzd θ̄kd z̄⎞⎠ .The real part of the Kähler metric is the Riemannian metric g. It can be checked thatthis metric has unit area. The Green’s function for this metric is known as the ArakelovGreen’s function and the following result is well known.Proposition 2.3. If M is a closed Riemann surface of genus H and if g is the canonicalmetric on M with unit area, then the Robin constant m(p) for g satisfiesm(p) = 2H − 2 + K (p)2π.Negative Mass Theorem for Surfaces of Positive Genus 1029Proof of Proposition 2.3. The Gaussian curvature K is given by2∂z∂z̄ logµ = −µK .The Arakelov Green’s function is given byG(z, w) = − 12πlog |E(z, w)| + 12H∑j,k=1(Im)−1jk Im(Z − W ) j Im(Z − W )k+m(z)2− logµ(z)8π+m(w)2− logµ(w)8π.Here, Z = I (z) and W = I (w) and E(z, w) is the prime form which plays the roleof z − w, is holomorphic in z and w, and transforms as a (−1/2,−1/2) form in eachvariable. We notice that log |E(z, w)| is harmonic, and so from (1.1) we have that forw = z,µ(z) = 4∂z∂z̄G(z, w) = Hµ(z) + 2∂z∂z̄m(z) + µ(z)K (z)4π.Hence we see that4µ−1∂z∂z̄m = 2 − 2H − K2π.	Theorem 1 is now an application of the following result on the mean field equationwhich is obtained from Theorem 1.2 of [DJLW] and its proof.Theorem 2.4. [DJLW] Let (M, g) be a closed surface of unit area and let h be a smoothpositive function on M. Suppose p0 is a point at which 8πm+2 log h attains its maximumvalue, and suppose in addition that log h(p0) < 8π − 2K (p0).Then the minimum of the functionalJ (u) = 116π∫M|∇u|2 d A +∫Mu d A − log∫Mheu d A (2.2)over functions u in the Sobolev space H1(M) is attained at a smooth function u satisfyingu = 8πheu − 8π. (2.3)Moreover, for this minimum point u we haveJ (u) < −(1 + logπ + maxp∈M(4πmg(p) + log h(p))). (2.4)Proof of Theorem 1. We take g to be the canonical metric on M , and we seth = e−4πmg .1030 K. OkikioluThen by Proposition 2.3, we have log h = −4πmg = 8π − 8πH − 2K < 8π − 2K .Hence we obtain the conclusion of Theorem 2.4 From (2.3), the function u satisfies∫Mheu d A = 1.From (2.2) and (2.3) we see thatJ (u) = 12∫Mu(heu + 1) d A.However, writingφ = u − 4πmg,we have∫Meφ d A = 1,and−1g eφ =18π(u −∫Mu). (2.5)Hence from (2.1),trace−1eφg= 18π∫Mu(heu + 1) = J (u)4π.Now from (2.4) and the fact thattrace−1S2,1= −1 − logπ4π,we see thattrace−1eφg< trace−1S2,1.In fact we remark that eφg minimizes trace−1 among unit area metrics in the conformalclass of g. Indeed, from [Ok] Theorem 1, we conclude that the minimum of trace−1eψ gamong conformal factors with∫eψ = 1, must in fact be attained at a metric eψg with∫M eψ d A = 1, which must also satisfy the Euler-Lagrange equation, namely that theRobin constant meψ g(p) is constant:−1g eψ =18π(ψ + 4πmg −∫M(ψ + 4πmg) d A).However, setting v = ψ + 4πmg , we find that v is a critical metric for J , and hencetrace−1eψ g= J (v)4π≥ J (u)4π= trace−1eφg.We also remark that in general the metric eφg need not coincide with the canonicalmetric or the constant curvature metric or the Arakelov metric, see [Ok2]. On a longthin rectangular torus, the minimizer is close to being a round sphere with a short wormhole joining the poles.Negative Mass Theorem for Surfaces of Positive Genus 1031Acknowledgement. I would like to thank Richard Wentworth for helpful discussions.Open Access This article is distributed under the terms of the Creative Commons Attribution NoncommercialLicense which permits any noncommercial use, distribution, and reproduction in any medium, provided theoriginal author(s) and source are credited.References[B] Beckner, W.: Sharp sobolev inequalities on the sphere and the moser-trudinger inequality. Annals ofMath. 138, 213–242 (1993)[CL] Carlen, E., Loss, M.: Competing symmetries, the logarithmic hls inequality and onofri’s inequalityon sn . Geom. Funct. Anal. 2, 90–104 (1992)[Ch] Chang, S.-Y.A.: Conformal invariants and partial differential equations. Bull. Amer. 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A negative mass theorem for surfaces of positive genus

A negative mass theorem for surfaces of positive genus

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