SUPPORTING SOBOLEV-TYPE INEQUALITIES

CSABA FARKAS, ALEXANDRU KRISTÁLY, AND ÁGNES MESTER

Abstract. Let (M, g) be an n-dimensional (n ≥ 3) compact Riemannian manifold
with Ric_{(M,g)}≥(n−1)g. If(M, g)supports an AB-type critical Sobolev inequality with
Sobolev constants close to the optimal ones corresponding to the standard unit sphere
(S^{n}, g0), we prove that (M, g) is topologically close to (S^{n}, g0). Moreover, the Sobolev
constants on(M, g)are precisely the optimal constants on the sphere(S^{n}, g0)if and only
if (M, g) is isometric to (S^{n}, g0); in particular, the latter result answers a question of
V.H. Nguyen.

1. Introduction

Let (M, g) be a smooth compact n-dimensional Riemannian manifold, n ≥ 3. The general theory of Sobolev inequalities shows that there exist A >0 and B >0such that

Z

M

|u|^{n−2}^{2n} dv_{g}
^{n−2}_{n}

≤A Z

M

|∇_{g}u|^{2}dv_{g} +B
Z

M

u^{2}dv_{g,} ∀u∈H_{1}^{2}(M). (1.1)
In fact, problem (1.1) is a part of the famous AB-program initiated by Aubin [1] concerning
the optimality of the constants A and B; for a systematic presentation of this topic, see
the monograph of Hebey [5, Chapters 4 & 5]. In particular, one can prove the existence
of B >0 such that (1.1) holds withA =A0 = _{n(n−2)}^{4} ω^{−}

2

nn, cf. [5, Theorem 4.6], the latter
value being the optimal Talenti constant in the Sobolev embedding H_{1}^{2}(R^{n}) ,→L^{2}^{∗}(R^{n}),
n ≥ 3, where 2^{∗} = 2n/(n − 2). Hereafter, ω_{n} = Vol_{g}_{0}(S^{n}) denotes the volume of the
standard unit sphere (S^{n}, g_{0}). If u ≡ 1 in (1.1), then we have B ≥ Vol_{g}(M)^{−}^{n}^{2}, where
Vol_{g}(S)denotes the volume of S ⊂M in (M, g). Moreover, if n ≥ 4 then the validity of
(1.1) withA=A_{0} = _{n(n−2)}^{4} ω^{−}

2

nn implies

B ≥ 1

n(n−1)ω^{−}

2

nnmax

M Scal(M,g),

where Scal_{(M,g)} is the scalar curvature of (M, g), cf. [5, Proposition 5.1].

In the model case when (M, g) = (S^{n}, g_{0}) is the standard unit sphere of R^{n+1}, Aubin
[1] proved that the optimal values of A and B in (1.1) are

A_{0} = 4

n(n−2)ω^{−}

2

nn and B_{0} =ω^{−}

2

nn, (1.2)

respectively; moreover, for every λ >1, the function u_{λ}(x) = (λ−cosd_{0}(x))^{1−}^{2}^{n}, x∈S^{n},
is extremal in (1.1), see also [5, Theorem 5.1]. Hereafter, d_{0}(x) = d_{S}^{n}(y_{0}, x), x ∈ S^{n},
where d_{S}^{n} denotes the standard metric on (S^{n}, g_{0}) and the element y_{0} ∈ S^{n} is arbitrarily
fixed. Note however that on the quotients M =S^{1}(r)×S^{2} ofS^{3} endowed with its natural

2010Mathematics Subject Classification. Primary: 58J05, 53C21, 53C24; Secondary: 46E35.

Key words and phrases. Riemannian geometry, compact manifold, rigidity, Sobolev inequality.

1

metric g (with r > 0 sufficiently small) inequality (1.1) is not valid for A = A_{0} and
B = Vol_{g}(M)^{−}^{n}^{2}, see [5, Proposition 5.7].

LetB_{M}(x, ρ)and B_{S}^{n}(y, ρ)be the open geodesic balls with radius ρ >0 and centers in
x∈M and y∈S^{n} in (M, g)and (S^{n}, g_{0}), respectively.

Our main result reads as follows:

Theorem 1.1. Let (M, g) be an n-dimensional (n ≥ 3) compact Riemannian manifold
with Ricci curvature Ric_{(M,g)} ≥(n−1)g and assume that the Sobolev inequality (1.1)holds
on (M, g) with some constants A, B >0. Then the following assertions hold:

(i) A≥A_{0} and B ≥B_{0}, where A_{0} and B_{0} are from (1.2);

(ii) there exists x_{0} ∈M such that for every y_{0} ∈S^{n} and ρ∈[0, π],
Volg(BM(x0, ρ))≥min

A_{0}
A,B_{0}

B
^{n}_{2}

Volg0(B_{S}^{n}(y0, ρ)). (1.3)
Remark 1.1. Note that (1.3) is valid on the whole [0,∞). Indeed, since the Ricci
curvature on (M, g) verifies Ric_{(M,g)} ≥ (n − 1)g, due to Bonnet-Myers theorem, the
diameter D_{M} := diam(M)of M is bounded from above byπ; accordingly, for everyρ≥π
one has B_{M}(x_{0}, ρ) = M and B_{S}^{n}(y_{0}, ρ) = S^{n}, thus (1.3) can be extended beyond π.

Perelman [10] states that for every n ≥ 2 there exists δ_{n} ∈ [0,1) such that if the n-
dimensional compact Riemannian manifold(M, g)with Ricci curvatureRic_{(M,g)} ≥(n−1)g
verifies Vol_{g}(M) ≥ (1−δ_{n})Vol_{g}_{0}(S^{n}), then M is homeomorphic to S^{n}; this result has
been improved by Cheeger and Colding [2, Theorem A.1.10] by replacing homeomorphic
to diffeomorphic. The latter result, the equality case in Bishop-Gromov inequality and
Theorem 1.1 imply the following topological rigidity for compact manifolds:

Corollary 1.1. Under the same assumptions as in Theorem 1.1, if max

A
A_{0}, B

B_{0}

≤(1−δ_{n})^{−}^{2}^{n},

then (M, g) is diffeomorphic to (S^{n}, g_{0}). Moreover, A = A_{0} and B = B_{0} if and only if
(M, g) is isometric to (S^{n}, g_{0}).

Remark 1.2. The statement of Corollary 1.1 is in the spirit of the results of Ledoux [9]

and do Carmo and Xia [4]. In these works certain Sobolev inequalities are considered on non-compact Riemannian manifolds with non-negative Ricci curvature, and the Riemann- ian manifold is isometric to the Euclidean space with the same dimension if and only if the Sobolev constants are precisely the Euclidean optimal constants. Further results in this direction can be found in the papers by Kristály [6, 7] and Kristály and Ohta [8].

Theorem 1.1 and Corollary1.1 seem to be the first contributions within this topic in the setting of compact Riemannian manifolds, answering also a question of Nguyen [11].

2. Proofs

Proof of Theorem 1.1. (i) The validity of the Sobolev inequality (1.1) on (M, g) and a
similar argument as in Hebey [5, Proposition 4.2] imply that A≥A_{0}.

By Remark 1.1, we have D_{M} :=diam(M) ≤ π. Since Ric_{(M,g)} ≥ (n − 1)g, by the
Bishop-Gromov comparison principle we have that for every x_{0} ∈ M and y_{0} ∈ S^{n}, the
function ρ7→ _{Vol}^{Vol}^{g}^{(B}^{M}^{(x}^{0}^{,ρ))}

g0(B_{S}n(y0,ρ)) is non-increasing on (0,∞); in particular, we have
1≥ Vol_{g}(B_{M}(x_{0}, ρ))

Vol_{g}_{0}(B_{S}^{n}(y_{0}, ρ)) ≥ Vol_{g}(B_{M}(x_{0}, π))

Vol_{g}_{0}(B_{S}^{n}(y_{0}, π)) = Vol_{g}(M)

Vol_{g}_{0}(S^{n}), ∀ρ∈[0, π]. (2.1)

Now, choosing u≡1 in (1.1), it follows that

B ≥Vol_{g}(M)^{−}^{n}^{2} ≥Vol_{g}_{0}(S^{n})^{−}^{n}^{2} =ω^{−}

2

nn =B_{0}.

(ii) If D_{M} = π, we have nothing to prove. Indeed, in this case (M, g) is isometric to
(S^{n}, g_{0}), see Cheng [3] and Shiohama [12], i.e., Vol_{g}(M) = Vol_{g}_{0}(S^{n}) and (2.1) implies at
once relation (1.3).

Accordingly, we assume that D_{M} < π. Fix x_{0},x˜_{0} ∈ M such that d_{g}(x_{0},x˜_{0}) = D_{M},
and y_{0} ∈ S^{n}. Let dv_{g} and dv_{g}_{0} be the canonical volume forms on (M, g) and (S^{n}, g_{0}),
respectively. Let f, s: (1,∞)→R be the functions defined as

f(λ) = Z

M

(λ−cosd_{g})^{2−n}dv_{g} and s(λ) =
Z

S^{n}

(λ−cosd_{0})^{2−n}dv_{S}^{n}, λ > 1, (2.2)
where d_{g} =d_{g}(x_{0},·) ans d_{0} =d_{S}^{n}(y_{0},·). It is easily seen that both functions f and s are
well-defined and smooth on (1,∞).

The proof will be provided in several steps.

Step 1 (local behavior of f and s around 1). We claim that lim inf

λ→1^{+}

f(λ)−λf^{0}(λ)

s(λ)−λs^{0}(λ) ≥1. (2.3)

By the layer cake representation of functions and a change of variables, we have that
I(λ) := f(λ)−λf^{0}(λ)

= Z

M

(λ−cosd_{g})^{1−n}((n−1)λ−cosd_{g})dv_{g}

= Z ∞

0

Vol_{g}({x∈M : (λ−cosd_{g})^{1−n}((n−1)λ−cosd_{g})> t})dt

= (n−2)
Z D_{M}

0

Vol_{g}(B_{M}(x_{0}, ρ))(λ−cosρ)^{−n}(nλ−cosρ) sinρdρ
+Vol_{g}(M)(λ−cosD_{M})^{1−n}((n−1)λ−cosD_{M}).

In a similar manner, we have
J(λ) := s(λ)−λs^{0}(λ)

= (n−2) Z π

0

Vol_{g}_{0}(B_{S}^{n}(y_{0}, ρ))(λ−cosρ)^{−n}(nλ−cosρ) sinρdρ
+Vol_{g}_{0}(S^{n})(λ+ 1)^{1−n}((n−1)λ+ 1).

Fix ε > 0 arbitrarily. Then the local behavior of the geodesic balls both on (M, g) and
(S^{n}, g_{0})implies that there exitsδ=δ_{ε}>0sufficiently small such that for everyρ∈(0, δ),

Vol_{g}(B_{M}(x_{0}, ρ))≥(1−ε)˜ω_{n}ρ^{n}
and

Vol_{g}_{0}(B_{S}^{n}(y_{0}, ρ))≤(1 +ε)˜ω_{n}ρ^{n},

where ω˜_{n} denotes the volume of the n-dimensional unit ball in R^{n}. Therefore, the above
estimates give that

I(λ) J(λ) ≥

(1−ε)(n−2)˜ω_{n}
Z δ

0

(λ−cosρ)^{−n}(nλ−cosρ)ρ^{n}sinρdρ
(1 +ε)(n−2)˜ωn

Z δ

0

(λ−cosρ)^{−n}(nλ−cosρ)ρ^{n}sinρdρ+ ˜s(λ, δ, n)

, (2.4)

where

˜

s(λ, δ, n) = (n−2) Z π

δ

Vol_{g}_{0}(B_{S}^{n}(y_{0}, ρ))(λ−cosρ)^{−n}(nλ−cosρ) sinρdρ
+Vol_{g}_{0}(S^{n})(λ+ 1)^{1−n}((n−1)λ+ 1).

Note first that s(λ, δ, n) =˜ O(1) asλ→1. Now, we show that

λ→1lim Z δ

0

(λ−cosρ)^{−n}(nλ−cosρ)ρ^{n}sinρdρ= +∞. (2.5)
Since cosρ >1−ρ^{2}, nλ−cosρ≥n−1and sinρ≥ _{π}^{2}ρ for every ρ∈(0, δ) and λ >1, it
suffices to prove that

λ→1lim Z δ

0

ρ^{n+1}

(λ−1 +ρ^{2})^{n}dρ= +∞.

In order to check the latter limit, by changes of variables one has

λ→1lim

(λ−1)^{n}^{2}^{−1}
Z δ

0

ρ^{n+1}

(λ−1 +ρ^{2})^{n}dρ

= lim

λ→1

Z δ/√ λ−1

0

τ^{n+1}

(1 +τ^{2})^{n}dτ [ρ=√

λ−1τ]

= Z ∞

0

τ^{n+1}
(1 +τ^{2})^{n}dτ

= 1 2

Z 1

0

θ^{n}^{2}(1−θ)^{n}^{2}^{−2}dθ

"

τ = r θ

1−θ

#

= 1 2 Bn

2 + 1,n 2 −1

.

Step 2 (ODE vs. ODI; global comparison of f and s). Due to Aubin [1], the extremal
function in (1.1) when(M, g) = (S^{n}, g_{0})isu_{λ}(x) = (λ−cosd_{0})^{1−}^{n}^{2} for everyλ >1. Thus,
inserting uλ into (1.1) when (M, g) = (S^{n}, g0) and using the notation in (2.2), we have
the following ODE:

s^{00}(λ)
(n−2)(n−1)

_{2}^{2}∗

= 2
nω^{−}

2

nn

1−λ^{2}

2(n−1)s^{00}(λ)−λs^{0}(λ) +s(λ)

, λ >1. (2.6)
LetK_{0} = _{n}^{2}ω^{−}

2

nn andC =K_{0}maxn

A
A0,_{B}^{B}

0

o

. Without loss of generality, we may assume
that A > A_{0}; indeed, sinceA≥ A_{0}, we may take A=A_{0}+ε for ε >0 sufficiently small.

Since B ≥Vol_{g}(M)^{−}^{n}^{2} ≥B_{0}, it turns out thatC > K_{0}. By introducing the function
H(λ) =

K_{0}
C

^{n}_{2}

J(s) =
K_{0}

C
^{n}_{2}

(s(λ)−λs^{0}(λ)),
one has H^{0}(λ) =−λ ^{K}_{C}^{0}^{n}_{2}

s^{00}(λ),therefore s^{00}(λ) =−^{H}^{0}_{λ}^{(λ)} ^{K}_{C}^{0}−^{n}_{2}

. This means that the second order ODE (2.6) is equivalent to the following first order ODE:

− H^{0}(λ)
λ(n−2)(n−1)

_{2}^{2}∗

=C

λ^{2}−1

2λ(n−1)H^{0}(λ) +H(λ)

, λ >1. (2.7)
Now, if we replace w_{λ}(x) = (λ−cosd_{g})^{1−}^{n}^{2} for every λ >1 into (1.1) and we explore
the eikonal equation |∇_{g}d_{g}|= 1 valid a.e. on M, we obtain

Z

M

(λ−cosd_{g})^{−n}dv_{g}
_{2}^{2}∗

≤A Z

M

(λ−cosd_{g})^{−n}sin^{2}d_{g}dv_{g}+B
Z

M

(λ−cosd_{g})^{2−n}dv_{g}.

By using the notation in (2.2), the latter inequality can be rewritten into
f^{00}(λ)

(n−2)(n−1)
_{2}^{2}∗

≤K_{0}
A

A_{0}

1−λ^{2}

2(n−1)f^{00}(λ)− A

A_{0}λf^{0}(λ) +

2−n 2

A
A_{0} +n

2
B
B_{0}

f(λ)

, for every λ >1. Since

1−λ^{2}

2(n−1)f^{00}(λ)−λf^{0}(λ) + 2−n

2 f(λ) = n−2 2

Z

M

(λ−cosd_{g})^{−n}sin^{2}d_{g}dv_{g} ≥0,
and C =K_{0}maxn

A
A0,_{B}^{B}

0

o

, the latter inequality implies that
f^{00}(λ)

(n−2)(n−1)
_{2}^{2}∗

≤C

1−λ^{2}

2(n−1)f^{00}(λ)−λf^{0}(λ) +f(λ)

, λ >1.

SinceI(λ) =f(λ)−λf^{0}(λ), we get the following first order ordinary differential inequality:

− I^{0}(λ)
λ(n−2)(n−1)

_{2}^{2}∗

≤C

λ^{2}−1

2λ(n−1)I^{0}(λ) +I(λ)

. (2.8)

We claim that

I(λ)≥H(λ), ∀λ >1. (2.9)

First of all, by (2.3) we clearly have that lim inf

λ→1^{+}

I(λ)

H(λ) = lim inf

λ→1^{+}

f(λ)−λf^{0}(λ)

K0

C

^{n}_{2}

(s(λ)−λs^{0}(λ))

≥ C

K_{0}
^{n}_{2}

>1.

Thus, for sufficiently small δ_{0} >0one has

I(λ)≥H(λ), ∀λ ∈(1, δ0+ 1).

Assume by contradiction that I(λ0) < H(λ0) for some λ0 >1. Clearly, λ0 >1 +δ0. Let us define

λ_{s}:= sup{λ < λ_{0} :I(λ) = H(λ)}< λ_{0}.
Thus for any λ∈[λ_{s}, λ_{0}] we have I(λ)≤H(λ).It is also clear that

− I^{0}(λ)

λ(n−2)(n−1) = f^{00}(λ)

(n−2)(n−1) >0 and

− H^{0}(λ)

λ(n−2)(n−1) = s^{00}(λ)

(n−2)(n−1) >0.

Let us define the increasing function ϕ_{λ} : (0,∞)→Rby
ϕ_{λ}(t) =t^{2}^{2}^{∗} +(n−2)

2 C(λ^{2}−1)t.

By relations (2.7), (2.8) and the definition of ϕλ, for every λ∈[λs, λ0] we have that ϕλ

− I^{0}(λ)
λ(n−2)(n−1)

=

− I^{0}(λ)
λ(n−2)(n−1)

_{2}^{2}∗

+(n−2)

2 C(λ^{2}−1)

− I^{0}(λ)
λ(n−2)(n−1)

≤CI(λ)

≤CH(λ)

=ϕ_{λ}

− H^{0}(λ)
λ(n−2)(n−1)

.

Therefore, the monotonicity of ϕ_{λ} implies

I^{0}(λ)≥H^{0}(λ), ∀λ ∈[λ_{s}, λ_{0}].

In particular λ 7→ I(λ)−H(λ) is non-decreasing on the interval [λ_{s}, λ_{0}]. Consequently,
we have

0 = I(λ_{s})−H(λ_{s})≤I(λ_{0})−H(λ_{0})<0,
a contradiction, which shows the validity of (2.9).

Step 3 (proving (1.3)). Due to (2.1), the claim is concluded once we prove
Vol_{g}(M)

Vol_{g}_{0}(S^{n}) ≥min
A_{0}

A ,B_{0}
B

^{n}_{2}

. (2.10)

Note that relation (2.9) is equivalent to (n−2)

Z DM

0

Volg(BM(x0, ρ)) nλ−cosρ

(λ−cosρ)^{n}sinρdρ+ Volg(M)(n−1)λ−cosD_{M}
(λ−cosD_{M})^{n−1}

≥ K0

C
^{n}_{2}

(n−2) Z π

0

Vol_{g}_{0}(B_{S}^{n}(y_{0}, ρ)) nλ−cosρ

(λ−cosρ)^{n}sinρdρ+ Vol_{g}_{0}(S^{n})(n−1)λ+ 1
(λ+ 1)^{n−1}

, for every λ >1.

Let us multiply the above inequality by λ^{n−2} and take the limit when λ → ∞; the
Lebesgue dominance theorem implies that both integrals tend to 0, remaining

Vol_{g}(M)≥
K_{0}

C
^{n}_{2}

Vol_{g}_{0}(S^{n}).

Since C = K_{0}maxn

A
A0,_{B}^{B}

0

o

, the latter relation implies (2.10) at once, which concludes

the proof of (1.3).

Proof of Corollary 1.1. Since max nA

A0,_{B}^{B}

0

o

≤ (1− δn)^{−}^{2}^{n}, by the quantitative volume
estimate (1.3) it follows that

Vol_{g}(M)≥(1−δ_{n})Vol_{g}_{0}(S^{n}).

The statement follows by Cheeger and Colding [2].

If (M, g) is isometric to (S^{n}, g_{0}), it is clear that A = A_{0} and B = B_{0}, due to Aubin
[1]. Conversely, when A =A_{0} and B =B_{0}, we apply (1.3) and (2.1) in order to obtain
Vol_{g}_{0}(B_{S}^{n}(y_{0}, ρ)) = Vol_{g}(B_{M}(x_{0}, ρ)) for every ρ ∈ [0, π] (in fact, for every ρ ∈ [0,∞)).

Now, the equality in the Bishop-Gromov comparison principle implies that (M, g)is iso-

metric to (S^{n}, g_{0}).

Acknowledgment. The authors are supported by the National Research, Develop- ment and Innovation Fund of Hungary, financed under the K_18 funding scheme, Project No. 127926. A. Kristály is also supported by the STAR-UBB grant.

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Department of Mathematics and Informatics, Sapientia University, Tg. Mureş, Roma- nia

E-mail address: farkas.csaba2008@gmail.com; farkascs@ms.sapientia.ro

Department of Economics, Babeş-Bolyai University, Cluj-Napoca, Romania, Institute of Applied Mathematics, Óbuda University, 1034 Budapest, Hungary

E-mail address: kristaly.alexandru@nik.uni-obuda.hu; alex.kristaly@econ.ubbcluj.ro Institute of Applied Mathematics, Óbuda University, 1034 Budapest, Hungary E-mail address: mester.agnes@yahoo.com