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**EXTRINSIC EIGENVALUES UPPER BOUNDS FOR** **SUBMANIFOLDS IN WEIGHTED MANIFOLDS**

### Fernando Manfio, Julien Roth, Abhitosh Upadhyay

**To cite this version:**

### Fernando Manfio, Julien Roth, Abhitosh Upadhyay. EXTRINSIC EIGENVALUES UPPER BOUNDS

### FOR SUBMANIFOLDS IN WEIGHTED MANIFOLDS. 2021. �hal-03334228�

SUBMANIFOLDS IN WEIGHTED MANIFOLDS

FERNANDO MANFIO, JULIEN ROTH, AND ABHITOSH UPADHYAY

Abstract. We prove Reilly-type upper bounds for divergence-type operators of the second order as well as for Steklov problems on submanifolds of Rie- mannian manifolds of bounded sectional curvature endowed with a weighted measure.

MSC 2010: 53C24, 53C42, 58J50.

Key words: Submanifolds, Reilly-type upper bounds, eigenvalues estimates, divergence- type operators, Steklov problems.

1. Introduction

Let (M^{n}, g) be ann-dimensional compact, connected, oriented manifold without
boundary, and consider an isometric immersionX :M^{n} →R^{n+1} in the Euclidean
space. The spectrum of Laplacian of (M, g) is an increasing sequence of real num-
bers

0 =λ_{0}(∆)< λ_{1}(∆)6λ_{2}(∆)6· · ·6λ_{k}(∆)6· · · −→+∞.

The eigenvalue 0 (corresponding to constant functions) is simple andλ_{1}(∆) is the
first positive eigenvalue. In [12], Reilly proved the following well-known upper
bound forλ_{1}(∆)

(1) λ_{1}(∆)6 n

V(M) Z

M

H^{2}dv_{g},

where H is the mean curvature of the immersion. He also proved an analogous inequality involving the higher order mean curvatures. Namely, forr∈ {1,· · ·, n}

(2) λ1(∆)

Z

M

Hr−1dvg

2

6V(M) Z

M

H_{r}^{2}dvg,

whereHr is ther-th mean curvature, defined by ther-th symmetric polynomial of the principal curvatures. Moreover, Reilly studied the equality cases and proved that equality in (1) or (2) is attained if and only ifX(M) is a geodesic sphere.

In the case of higher codimension, Reilly also proved that

(3) λ1(∆)6 n

V(M) Z

M

kHk^{2}dvg,

where H is here the mean curvature vector, with equality if and only if M is minimally immersed in a geodesic sphere.

The first author is supported by Fapesp, grant 2019/23370-4.

1

The Reilly inequality can be easily extended to submanifolds of the sphere S^{n}
using the canonical embedding ofS^{n} into R^{n+1}:

(4) λ1(∆)6 n

V(M) Z

M

(kHk^{2}+ 1)dvg.

Moreover, El Soufi and Ilias [7] proved an analogue for submanifold of the hyperbolic space as

(5) λ_{1}(∆)6 n

V(M) Z

M

(kHk^{2}−1)dv_{g}.

In case the ambient space has non-constant sectional curvature, Heintze [10] proved the following weaker inequality

(6) λ_{1}(∆)6n(kHk^{2}+δ),

where the ambient sectional curvature is bounded above byδ.

On the other hand, more recently, in [14], the second author prove the following general inequality

(7) λ1(LT ,f) Z

M

tr (S)µf

^{2}
6

Z

M

tr (T)µf

Z

M

kHSk^{2}+kS∇fk^{2}
µf,
where µf = e^{−f}dvg is the weighted measure of (M, g) endowed with the density
e^{−f},T, Sare two symmetric, free-divergence (1,1)-tensors withT positive definite,
andLT ,f is the second order differential operator defined for any smooth function
uonM by

LT ,f =−div(T∇u) +h∇f, T∇ui.

Whenf = 0 and the tensor S andT are associated with higher order mean cur- vatures Hs and Hr, we recover the inequality of Alias and Malacarn´e [2] and, in particular, Reilly’s inequality (2) ifr= 0.

The first result of this paper gives upper bounds for the first eigenvalue of the operator LT ,f for submanifolds of Riemannian manifolds with sectional curvature bounded by above which generalizes inequality (7) in the non-constant curvature case. Namely, we prove the following.

Theorem 1.1. Let( ¯M^{n+p},g,¯ µ¯f)be a weighted Riemannian manifold with sectional
curvaturesectM¯ 6δandµ¯f =e^{−f}dvg¯. Let(M, g)be a closed Riemannian manifold
isometrically immersed into ( ¯M^{n+p},g) by¯ X. We endow M with the weighted
measureµf =e^{−f}dvg. Let T be a positive definite (1,1)-tensor on M and denote
byλ1 the first positive eigenvalue of the operatorLT ,f.

(1) If δ60, then λ16sup

M

δtr (T) + sup

M

kHT−T∇fk tr (S)

kHS−S∇fk

.
(2) If δ >0 andX(M)is contained in a geodesic ball of radius ^{π}

4√ δ,

λ_{1}6
Z

M

tr (T)µ_{f}

Vf(M)

δ+

Z

M

kHS−S(∇f)k^{2}µf

Vf(M) inf (tr (S)^{2})

.

The second eigenvalue problem that we consider in this paper is the Steklov problem associated with the operator LT ,f on a submanifold Ω with non-empty boundary∂Ω =M of a Riemannian manifold with sectional curvature bounded by above. We can consider the following generalized weighted Steklov problem

(8)

LT ,fu= 0 on Ω,

∂u

∂νT

=σu onM =∂Ω,
where _{∂ν}^{∂u}

T =hT(∇u), νi. In the case wheree f is constant, the operatorLT ,f is of
particular interest for the study ofr-stability whenT =T_{r}is the tensor associated
withr-th mean curvature (see [1] for instance). More precisely, from [3], we know
that this problem (8) has a discrete nonnegative spectrum and we denote by σ1

its first eigenvalue. In [13], the second author has obtained upper bounds for this problem for domains of a manifold lying in a Euclidean space. Namely, he has proved

σ_{1}
Z

M

tr (S)µe_{f}
2

6 Z

Ω

tr (T)µ_{f}
Z

M

||HS||^{2}+||S∇f||^{2}
eµ_{f},
where µ_{f} = e^{−f}dv_{g} and µe_{f} = e^{−f}dv

eg are respectively, the weighted measures
of (Ω, g) and (M,eg) endowed with the density e^{−f} and where T and S are two
symmetric, free-divergence (1,1)-tensors withTpositive definite. Note that without
density and forT = Id , this inequality has been proven by Ilias and Makhoul [11].

The second result of the present paper gives a generalization of this estimate when the manifold with boundary (M, g) is immersed into an ambient Riemannian manifold of sectional curvature bounded by above. Namely, we prove the following.

Theorem 1.2. Let( ¯M^{n+p},g,¯ µ¯f)be a weighted Riemannian manifold with sectional
curvature sectM¯ 6 δ and µ¯_{f} = e^{−f}dv_{¯}_{g}. Let (Ω, g) be a compact Riemannian
manifold with non-empty boundary M isometrically immersed into ( ¯M^{n+p},¯g) by
X. We endowΩand M, respectively with the weighted measureµ_{f} =e^{−f}dv_{g} and

˜

µf = e^{−f}dv˜g, where ˜g is the induced metric on M. Let T, S be a symmetric,
divergence-free and positive definite (1,1)-tensors on Ω and M, respectively, and
denote byσ1 the first eigenvalue of the Steklov problem (8).

(1) If δ6 0 and X(Ω) is contained in the geodesic ball B(p, R) or radius R, wherepis the center of mass ofM for the measureµef, then

σ_{1}6sup

Ω

δtr (T) + sup

Ω

kH_{T} −T(∇f)k
tr (T)

kH_{T} −T(∇f)k

×

δ+sup_{M}kHS−S(∇f)k^{2}
infM(tr (S))^{2}

Vf(Ω)
Vf(M)s^{2}_{δ}(R).

(2) If δ >0 andX(Ω) is contained in a geodesic ball of radius ^{π}

4√ δ, then

σ_{1}6
Z

Ω

tr (T)µf

Vf(M)

δ+

Z

M

kHS−S(∇f)k^{2}µef

Vf(M) inf (tr (S)^{2})

.

Finally, we will consider the so-called eigenvalue problem for Wentzell boundary conditions

(SW)

( ∆u = 0 in Ω

−b∆ue −∂u

∂ν = αu onM ,

wherebis a given positive constant, Ω is a submanifold with non-empty boundary

∂Ω =M of a Riemannian manifoldM with sectional curvature bounded by above,
and ∆,∆ denote the Laplacians on Ω ande M, respectively. It is clear that ifb= 0,
then we recover the classical Steklov problem. The spectrum of this problem is
an increasing sequence (see [5]) with 0 as first eigenvalue which is simple and the
corresponding eigenfunctions are the constant ones. We denote by α1 the first
positive eigenvalue. In [14], the second author proved the following estimate when
Ω is a submanifold of the Euclidean spaceR^{n}

α1

Z

∂M

tr (S)dvg

^{2}
6

nV(M) +b(n−1)V(∂M)Z

∂M

kHSk^{2}dvg

.

In the following theorem, we obtain a comparable estimate when the ambient space is of bounded sectional curvature. Namely, we prove

Theorem 1.3. Let ( ¯M^{n+p},g)¯ be a Riemannian manifold with sectional curvature
sectM¯ 6δ. Let(Ω, g)be a compact Riemannian manifold with non-empty boundary
M isometrically immersed into( ¯M^{n+p},¯g)byX. We denote byegthe induced metric
on M. Let S be a symmetric, divergence-free and positive definite (1,1)-tensor on
M and denote by α1 the first eigenvalue of the Steklov-Wentzell problem (SW).

(1) If δ6 0 and X(Ω) is contained in the geodesic ball B(p, R) or radius R, wherepis the center of mass ofM, then

α16h nV(Ω)

V(M)+b(n−1)−δs^{2}_{δ}(R)

V(Ω) V(M)+b

i

δ+ sup_{M}kHSk^{2}
infM(tr (S))^{2}

.
(2) If δ >0 and soX(Ω) is contained in a geodesic ball of radius ^{π}

4√ δ, then

α_{1}6

nV(Ω)

V(M)+b(n−1)

δ+

Z

M

kHSk^{2}dv

eg

V(M) inf (tr (S)^{2})

.

2. Preliminaries

Let ( ¯M^{n+p},¯g,µ¯f) be a weighted Riemannian manifold with sectional curvature
sectM¯ 6δand weighted measure ¯µf =e^{−f}dv¯g. Letpa fixed point in ¯M, we denote
byr(x) the geodesic distance betweenxandp. Moreover, we define the vector field
X byX(x) :=sδ(r(x))( ¯∇r)(x),sδ is the function defined by

sδ(r) =

√1 δ sin(√

δr) if δ >0

r if δ= 0

√1

|δ|sinh(p

|δ|r) if δ <0.

We also define

cδ(r) =

cos(√

δr) if δ >0

1 if δ= 0

cosh(p

|δ|r) if δ <0.

Hence, we have

c^{2}_{δ}+δs^{2}_{δ} = 1, s^{0}_{δ}=c_{δ} and c^{0}_{δ}=−δs_{δ}.

In addition, let (M^{n}, g) be a closed Riemannian manifold isometrically immersed
into ( ¯M^{n+p},¯g) byφ. Ifδ >0, then we assume thatφ(M) is contained in a geodesic
ball of radius ^{π}

2√

δ. We endowM with the weighted measureµf =e^{−f}dvg. We can
define onM a divergence associated with the volume formµf =e^{−f}dvg by

divfY = divY − h∇f, Yi or, equivalently,

d(ι_{Y}µ_{f}) = div_{f}(Y)µ_{f},

where ∇ is the gradient on Σ, that is, the projection on TΣ of the gradient ¯∇ on ¯M. We call it the f-divergence. We recall briefly some basic facts about the f-divergence. In the case where Σ is closed, we first have the weighted version of the divergence theorem:

(9)

Z

Σ

divfY µf = 0,

for any vector fieldY on Σ. From this, we deduce easily the integration by parts formula

(10)

Z

Σ

udivfY µf =− Z

Σ

h∇u, Yiµf,

for any smooth functionuand any vector fieldY on Σ. First, we prove the following elementary lemma which generalize in non constant curvature the classical Hsiung- Minkowski formula (see [8, 13] for instance).

Lemma 2.1. Let T be a symmetric divergence-free positive (1,1)-tensor on M. Then the following hold

(1) divf T X^{>}

>tr (T)cδ+hX, HT −T(∇f)i.

(2) Z

M

tr (T)cδµf 6− Z

M

hX, HT−T(∇f)iµf. (3) δ

Z

M

hT X^{>}, X^{>}iµf >

Z

M

tr (T)c^{2}_{δ}µ_{f}−
Z

M

kHT −T(∇f)ksδc_{δ}µ_{f}.

Proof. The proof is a straightforward consequence of the analogue non-weighted result proven by Grosjean. Namely, in [8], the author has shown that

div_{M}(T X^{>})>tr (T)c_{δ}(r) +hX, H_{T}i.

Hence, from the definition of thef-divergence, we have
divf(T X^{>}) = div(T X^{>})− h∇f, T X^{>}i

> tr (T)c_{δ}(r) +hX, HT −T(∇f)i,

and this proves part (1). For the second part, we integrate the last inequality with
respect to the measureµ_{f} and we get immediately

Z

M

tr (T)c_{δ}µ_{f} 6−
Z

M

hX, H_{T}−T(∇f)iµ_{f},
since

Z

M

divf(T X^{>})µf = 0.

Finally, for the last part, ifδ= 0, then cδ = 1 and we get directly the conclusion from the second one by using

|hX, H_{T} −T(∇f)i|6||X|| · ||H_{T} −T(∇f)||=s_{δ}||H_{T} −T(∇f)||.

Ifδ6= 0, sinceX^{>}=sδ(r)∇r=−δ∇cδ(r), then we have
δ

Z

M

hT X^{>}, X^{>}iµ_{f} = 1
δ

Z

M

hT(∇c_{δ},∇c_{δ})µ_{f}

= −1 δ Z

M

divf(T∇cδ)cδµf

= Z

M

divf(X^{>})cδµf

>

Z

M

tr (T)c^{2}_{δ}µf−
Z

M

kHT −T(∇f)ksδcδµf, where we have used the first part of the lemma and the well-known Cuachy-Schwarz

inequality.

3. Two key lemma

In this section we will prove two basic key lemma that will be used throughout the paper.

Lemma 3.1. Let ( ¯M^{n+p},¯g,µ¯f)be a weighted Riemannian manifold with sectional
curvature sectM¯ 6δ 60, and µ¯f =e^{−f}dvg¯. Let (M, g) be a closed Riemannian
manifold isometrically immersed into( ¯M^{n+p},g), and we endow¯ M with the weighted
measureµ_{f} =e^{−f}dv_{g}. Let S be a symmetric, divergence-free and positive definite
(1,1)-tensor on M. Then, we have

Z

M

kXk^{2}µf

V_{f}(M) > 1

δ+kHS−S(∇f)k^{2}_{∞}
inf(tr (S))^{2}

.

Proof. We have Z

M

tr (S)−δhSX^{>}, X^{>}i
µf=

Z

M

tr (S)−div(SX^{>})cδ(r)
µf

6 Z

M

tr (S)−tr (S)c^{2}_{δ}(r)− hHS−S(∇f), Xicδ(r)
µf

6 Z

M

δtr (S)s^{2}_{δ}(r)− hHS−S(∇f), Xicδ(r)
µf

6 Z

M

δtr (S)s^{2}_{δ}(r) +kHS−S(∇f)k_{∞}
inf(tr (S))

Z

M

tr (S)s_{δ}(r)c_{δ}(r)

µ_{f}
6 kHS−S(∇f)k∞

inf(tr (S)) Z

M

sδ(r)div(SX^{>})−sδ(r)hHS−S(∇f), Xi
µf

+δinf(tr (S))R

Ms^{2}_{δ}(r)µf

6−kHS−S(∇f)k_{∞}
inf(tr (S))

Z

M

cδ(r)sδ(r)hS∇^{M}r,∇^{M}ri+s^{2}_{δ}(r)hHS−S(∇f),∇^{M}ri
µf

+δinf(tr (S))R

Ms^{2}_{δ}(r)µf

6

δinf(tr (S)) +kHS−S(∇f)k^{2}_{∞}
inf(tr (S))

Z

M

s^{2}_{δ}(r)µf

−kHS−S(∇f)k_{∞}
inf(tr (S))

Z

M

cδ(r)sδ(r)hS∇^{M}r,∇^{M}riµf.
Hence, we get

Z

M

tr (S)µ_{f} 6

δinf(tr (S)) +kHS−S(∇f)k^{2}_{∞}
inf(tr (S))

Z

M

s^{2}_{δ}(r)µ_{f}

−kHS−S(∇f)k∞

inf(tr (S)) Z

M

cδ(r)sδ(r)hS∇^{M}r,∇^{M}riµf

+δ Z

M

s^{2}_{δ}(r)hS∇^{M}r,∇^{M}riµf

6

δinf(tr (S)) +kHS−S(∇f)k^{2}_{∞}
inf(tr (S))

Z

M

s^{2}_{δ}(r)µf

+ Z

M

δs^{2}_{δ}(r)−cδ(r)sδ(r)kHS−S(∇f)k∞

inf(tr (S))

hS∇^{M}r,∇^{M}riµf.

Sinceδ60, the second term of the right hand side is nonpositive, and thus we get
inf(tr (S))V_{f}(M) 6

Z

M

tr (S)µ_{f}
6

δinf(tr (S)) +kH_{S}−S(∇f)k^{2}_{∞}
inf(tr (S))

Z

M

s^{2}_{δ}(r)µ_{f},
which gives immediately the result sincekXk=s_{δ}(r).

Lemma 3.2. Let ( ¯M^{n+p},¯g,µ¯f)be a weighted Riemannian manifold with sectional
curvature sectM¯ 6 δ, with δ > 0, and µ¯f = e^{−f}dv¯g. Let (M, g) be a closed
Riemannian manifold isometrically immersed into ( ¯M^{n+p},g)¯ by X so that X(M)
is contained in a geodesic ball of radius ^{π}

2√

δ. We endow M with the weighted
measureµf =e^{−f}dvg. Let S be a symmetric, divergence-free and positive definite
(1,1)-tensor on M. Then, we have

1−

Z

M

c_{δ}(r)µ_{f}
Vf(M)

2

> 1

1 + Z

M

kHS−S(∇f)k^{2}µf

δinf(tr (S))^{2}V_{f}(M)
.

Proof. For a sake of compactness, we will write

α= Z

M

c_{δ}(r)µ_{f}

Vf(M) and β = 1 + Z

M

kHS−S(∇f)k^{2}µ_{f}
δinf(tr (S))^{2}Vf(M) .

We thus have to show that (1−α^{2})β >1. We have

(1−α^{2})β = β−

Z

M

cδ(r)µf

V_{f}(M)

2

−

Z

M

cδ(r)µf

V_{f}(M)

2Z

M

kHS−S(∇f)k^{2}µf

δinf(tr (S))^{2}V_{f}(M)

> β−

Z

M

tr (S)c_{δ}(r)µ_{f}
inf(tr (S))V_{f}(M)

2

−

Z

M

c_{δ}(r)^{2}µ_{f}
V_{f}(M)

Z

M

kHS−S(∇f)k^{2}µ_{f}
δinf(tr (S))^{2}V_{f}(M)

> β−

Z

M

s_{δ}(r)kHS−S(∇f)kµf

inf(tr (S))Vf(M)

2

−

Z

M

c_{δ}(r)^{2}µ_{f}
Vf(M)

Z

M

kHS−S(∇f)k^{2}µ_{f}
δinf(tr (S))^{2}Vf(M)

> β− Z

M

s^{2}_{δ}(r)µf

Z

M

kHS−S(∇f)k^{2}µf

inf(tr (S))^{2}Vf(M)^{2}

−

Z

M

cδ(r)^{2}µf

V_{f}(M)

Z

M

kHS−S(∇f)k^{2}µf

δinf(tr (S))^{2}V_{f}(M)

> β− Z

M

kHS−S(∇f)k^{2}µ_{f}
δinf(tr (S))^{2}Vf(M)^{2}

Z

M

(s^{2}_{δ}(r) +δ^{2}c^{2}_{δ}(r))µf

= β− Z

M

kHS−S(∇f)k^{2}µf

δinf(tr (S))^{2}V_{f}(M)

= 1,

and this concludes the proof.

4. Proofs of the Theorems

Proof of Theorem 1.1. Caseδ60. Letp∈M be a fixed point and let{x1,· · ·, xN} be the normal coordinates ofM centered atp. For anyx∈M,r(x) is the geodesic distance betweenpandxoverM. We want to use as test functions the functions

s_{δ}(r)
r xi,

for 16i6N. For this purpose, we will choosepas the center of mass ofM with respect for the measureµf, that is,pis the only point inM so that

Z

M

sδ(r)

r x_{i}µ_{f} = 0,

for anyi∈ {1,· · · , N}. Note at that point that we assume that M is contained in a ball of radius π

4√

δ whenδis positive allows us to ensure thatM is contained in a ball of radius π

4√

δ centered atp. This holds also for Theorems 1.2 and 1.3. These

functions are candidates for test functions since they areL^{2}(µf)-orthogonal to the
constant functions which are the eigenfunctions for the first eigenvalue λ0 = 0.

Thus, we have (11) λ1

Z

M N

X

i=1

s^{2}_{δ}(r)
r^{2} x^{2}_{i}µf 6

Z

M N

X

i=1

T∇

sδ(r) r xi

,∇

sδ(r) r xi

µf.

We recall that Grosjean proved in [8, Lemma 2.1] that (12)

N

X

i=1

T∇

s_{δ}(r)
r x_{i}

,∇

s_{δ}(r)
r x_{i}

6tr (T)−δ

T X^{>}, X^{>}

. Moreover, by Lemma 2.1, item (3), we have

δ Z

M

hT X^{>}, X^{>}iµf >

Z

M

tr (T)c^{2}_{δ}(r)µf−
Z

M

kHT −T(∇f)ksδ(r)cδ(r)µf

which together with (11) and (12) gives
λ_{1}

Z

M

s^{2}_{δ}(r)µ_{f} 6
Z

M

tr (T)−c^{2}_{δ}(r)tr (T) +kH_{T} −T(∇f)ks_{δ}(r)c_{δ}(r)
µ_{f}
6 δ

Z

M

s^{2}_{δ}(r)tr (T)µ_{f}+
Z

M

kH_{T}−T(∇f)ks_{δ}(r)c_{δ}(r)µ_{f},
where we have usedc^{2}_{δ}+δs^{2}_{δ} = 1. Hence, we obtain

λ1

Z

M

s^{2}_{δ}(r)µf 6 δ
Z

M

s^{2}_{δ}(r)tr (T)µf

(13)

+ sup

M

kHT−T(∇f)k tr (S)

Z

M

tr (S)sδ(r)cδ(r)µf. Now, we claim the following

Lemma 4.1. We have Z

M

tr (S)sδ(r)cδ(r)µf 6 Z

M

kHS−S(∇f)ks^{2}_{δ}(r)µf.
Reporting this into (13), we get

λ1

Z

M

s^{2}_{δ}(r)µf 6
Z

M

"

δtr (T) + sup

M

kHT −T(∇f)k tr (S)

kHS−S(∇f)k

#

s^{2}_{δ}(r)µf.
which gives immediately the desired estimate

λ16sup

M

δtr (T) + sup

M

kHT−T∇fk tr (S)

kHS−S∇fk

.

This concludes the proof for the case δ <0, up to the proof of the lemma that we give now.

Proof of Lemma 4.1. Multiplying the first part of Lemma 2.1 for the tensorS by sδ(r), we get

div_{f}(SX^{>})s_{δ}(r)>tr (S)c_{δ}(r)s_{δ}(r)− hX, H_{S}−S(∇f)i,
and thus, by the Cauchy-Schwarz inequality, we have

div_{f}(SX^{>})s_{δ}(r)>tr (S)c_{δ}(r)s_{δ}(r)− kHS−S(∇f)ksδ(r).

Now, integrating this relation and using the integration by parts in the formula (10), we get

Z

M

tr (S)c_{δ}(r)s_{δ}(r)µ_{f}−
Z

M

kH_{S}−S(∇f)ks_{δ}(r)µ_{f} 6 −
Z

M

h∇s_{δ}(r), SX^{>}iµ_{f}
6 −

Z

M

c_{δ}(r)s_{δ}(r)h∇r, S∇i
6 0,

sinceS is positive. This concludes the proof of Lemma 4.1.

Caseδ >0. Like in the caseδ <0, we use ^{s}^{δ}_{r}^{(r)}xi, 16i6N, as test functions.

Using (12) again, we get

(14) λ1

Z

M

s^{2}_{δ}(r)µf 6
Z

M

tr (T)−δ

T X^{>}, X^{>}

µf.

On the other, we use another test function in this case, namely c_{δ}(r)−c_{δ} where
for more convenience, we have denoted by cδ the mean value of cδ(r), that is
cδ = 1

V_{f}(M)
R

Mcδ(r)µf. Here again, this function is L^{2}(µf)-orthogonal to the
constant functions, so it is a candidate for being a test function. Hence, we have

λ_{1}
Z

M

(c_{δ}(r)−c_{δ})^{2}µ_{f} 6
Z

M

hT∇(c_{δ}(r)−c_{δ}),∇(c_{δ}(r)−c_{δ})iµ_{f}
6

Z

M

hT∇c_{δ}(r),∇c_{δ}(r)iµ_{f}
6 δ^{2}

Z

M

s^{2}_{δ}(r)hT∇r,∇riµ_{f}
6 δ^{2}

Z

M

T X^{>}, X^{>}

µf. From this, we deduce immediately that

(15) λ1

Z

M

c^{2}_{δ}(r)µf 6δ^{2}
Z

M

T X^{>}, X^{>}

+ λ1

V_{f}(M)
Z

M

cδ(r)µf

^{2}
.
Now, using the fact thatc^{2}_{δ}+δs^{2}_{δ}= 1, (15) plusδtimes (14) gives

λ1Vf(M)6δ Z

M

tr (T)µf+

Z

M

c_{δ}(r)µ_{f}
V_{f}(M)

2

and thus

λ1Vf(M) 1− R

Mc_{δ}(r)µ_{f}
Vf(M)

^{2}!
6δ

Z

M

tr (T)µf. Now, we conclude by using Lemma 3.2 to get the desired upper bound

λ_{1}6
Z

M

tr (T)µ_{f}

Vf(M)

δ+

Z

M

kHS−S(∇f)k^{2}µf

Vf(M) inf (tr (S)^{2})

,

and this concludes the proof of Theorem 1.1.

Proof of Theorem 1.2. Case δ 6 0. Like in the proof of Theorem 1.1, we will considerp∈M as the center of mass of Ω,{x1, . . . , xN} the normal coordinates of M centered atp and byr(x) the geodesic distance (on M) between xand p, for anyx∈M. By the choice of p, we have

Z

M

s_{δ}(r)

r xieµf = 0,

for anyi∈ {1, . . . , N}, and we can use the functions ^{s}^{δ}_{r}^{(r)}xias test functions in the
variational characterization ofσ1. Thus, we have

σ1

Z

M N

X

i=1

s^{2}_{δ}(r)
r^{2} x^{2}_{i}µef 6

Z

Ω N

X

i=1

T∇

sδ(r) r xi

,∇

sδ(r) r xi

µf

6 Z

Ω

tr (T)−δ

T X^{>}, X^{>}

µf, (16)

where we have used inequality (12) to get the second line. Here,X=s_{δ}(r)∇rand
X^{>} =sδ(r)∇r is the tangent component ofX to Ω. On the other hand, by item
(3) of Lemma 2.1 applied toM, we have

δ Z

Ω

hT X^{>}, X^{>}iµf >

Z

Ω

tr (T)c^{2}_{δ}µf−
Z

Ω

kHT −T(∇f)ksδcδµf. Reporting into (16), we have

σ1

Z

M

s^{2}_{δ}(r)µef 6
Z

Ω

tr (T)−c^{2}_{δ}(r)tr (T) +kHT −T(∇f)ksδ(r)cδ(r)
µf

6 δ Z

Ω

s^{2}_{δ}(r)tr (T)µf+
Z

Ω

kHT −T(∇f)ksδ(r)cδ(r)µf, 6 δ

Z

Ω

s^{2}_{δ}(r)tr (T)µf+ sup

Ω

kHT −T(∇f)k tr (T)

Z

Ω

tr (T)sδ(r)cδ(r)µf. From Lemma 4.1 applied on Ω for the tensorT, we have

Z

Ω

tr (T)s_{δ}(r)c_{δ}(r)µ_{f} 6
Z

Ω

kHT−T(∇f)ks^{2}_{δ}(r)µ_{f},
which yields to

σ1

Z

M

s^{2}_{δ}(r)µef 6
Z

Ω

s^{2}_{δ}(r)

δtr (T) + sup

Ω

kHT −T(∇f)k tr (T)

kHT −T(∇f)k

µf. Moreover, from the assumption that Ω is contained in the ball of radius B(p, R), we get thatsδ(r)6sδ(R) and thus

σ_{1}
Z

M

s^{2}_{δ}(r)µe_{f} 6s^{2}_{δ}(R)V_{f}(Ω) sup

Ω

δtr (T) + sup

Ω

kHT−T(∇f)k tr (T)

tkH_{T} −T(∇f)k

. Finally, by Lemma 3.2, we have

Z

M

sδ(r)^{2}eµf

V_{f}(M) > 1

δ+sup_{M}kHS−S(∇f)ke ^{2}
inf_{M}(tr (S))^{2}

,

which give the desired estimate whenδ60:

σ_{1}6sup

Ω

δtr (T) + sup

Ω

kH_{T} −T(∇f)k
tr (T)

kHT −T(∇f)k

×

"

δ+sup_{M}kHS−S(∇fe )k^{2}
infM(tr (S))^{2}

# Vf(Ω)
Vf(M)s^{2}_{δ}(R).

Case δ >0. Like in the caseδ60, we have

(17) σ_{1}

Z

M

s^{2}_{δ}(r)eµ_{f} 6
Z

Ω

tr (T)−δ

T X^{>}, X^{>}

µ_{f}

by using ^{s}^{δ}_{r}^{(r)}xi, 16i6N as test functions. In addition, we also use another test
function,c_{δ}(r)−ce_{δ}, withce_{δ} = 1

Vf(M) R

Mc_{δ}(r)eµ_{f}. By a computation analogue to
the proof of Theorem 1.1, we have

σ_{1}
Z

M

(c_{δ}(r)−ce_{δ})^{2}eµ_{f} 6
Z

Ω

hT∇(c_{δ}(r)−ce_{δ}),∇(c_{δ}(r)−ce_{δ})iµ_{f}
6

Z

Ω

hT∇cδ(r),∇cδ(r)iµ_{f}
6 δ^{2}

Z

Ω

s^{2}_{δ}(r)hT∇r,∇riµf

6 δ^{2}
Z

Ω

T X^{>}, X^{>}

µf. From this, we deduce

(18) σ1

Z

M

c^{2}_{δ}(r)µef 6δ^{2}
Z

Ω

T X^{>}, X^{>}

µf + σ1

Vf(M) Z

M

cδ(r)eµf

^{2}
.
Now, using the fact thatc^{2}_{δ}+δs^{2}_{δ}= 1, (18) plusδtimes (17) gives

σ1Vf(M)6δ Z

Ω

tr (T)µf+

Z

M

c_{δ}(r)eµ_{f}
Vf(M)

2

and so

σ_{1}V_{f}(M) 1−
R

Mc_{δ}(r)µe_{f}
Vf(M)

^{2}!
6δ

Z

Ω

tr (T)µ_{f}.
Finally, since we have

1−

Z

M

cδ(r)µef

Vf(M)

2

> 1

1 + Z

M

kHS−S(∇fe )k^{2}µef

δinf(tr (S))^{2}Vf(M)
,

by Lemma 3.2, we get

σ_{1}6
Z

Ω

tr (T)µf

Vf(M)

δ+

Z

M

kHS−S(∇fe )k^{2}µef

Vf(M) inf (tr (S))^{2}

,

and this concludes the proof.

Proof of Theorem 1.3. Here again we consider differently the two cases. Caseδ60.

First, we recall the variational characterization ofα1 (see [5])

(19) α_{1}= inf

Z

Ω

k∇uk^{2}dvg+b
Z

M

k∇uk^{2}dv

eg

Z

M

u^{2}dv

eg

Z

M

udv

eg= 0

.

As in the proof of Theorem 1.2, we use sδ(r)

r xias test functions, whereris the geo- desic distance to the center of masspof Ω and{x1,· · ·, xN}the normal coordinates ofM centered atp. Hence, we have

α_{1}
Z

M N

X

i=1

s^{2}_{δ}(r)
r^{2} x^{2}_{i}dv

eg6 Z

Ω N

X

i=1

∇
s_{δ}(r)

r x_{i}

dv_{g}+b
Z

M N

X

i=1

∇
s_{δ}(r)

r x_{i}

dv

eg

that is,

α1

Z

M

sδ(r)^{2}dv

eg6 Z

Ω

n−δkX^{>}k^{2}
dvg

+b Z

M

(n−1)−δkX^{>}^{e}k^{2}
dveg,
(20)

where we use use inequality (12) twice, once on Ω for the first term and once on
M for the second term. Moreover we have, kX^{>}^{e}k 6 kX^{>}k 6 kXk =s_{δ}(r) and
sinceδ is nonpositive, s_{δ} is an increasing function. So, by the assumption that Ω
is contained in the ballB(p, R), we have

α1

Z

M

s^{2}_{δ}(r)dv

eg6 n−δsδ(R)^{2}

V(Ω) +b(n−1)−δsδ(R))V(M) 6

nV(Ω) +b(n−1)V(M)

−δsδ(R)^{2}

V(Ω) +bV(M) . (21)

We conclude by applying Lemma 3.2, which says forf = 0, Z

M

sδ(r)^{2}dv

eg

V(M) > 1
δ+ sup_{M}kHSk

inf_{M}(tr (S))^{2}
,

to obtain the desired estimate whenδ60, that is,
α_{1}6h

nV(Ω)

V(M)+b(n−1)−δs^{2}_{δ}(R)

V(Ω) V(M)+b

i

δ+ sup_{M}kHSk^{2}
inf_{M}(tr (S))^{2}

.

Case δ >0. Like in the cas δ60, using the functions s_{δ}(r)

r xi as test functions in the variational characterization ofα1, we get

(22) α1

Z

M

s^{2}_{δ}(r)dv

eg6 Z

Ω

n−δkX^{>}k^{2}
dvg+b

Z

M

(n−1)−δkX^{>}^{e}k^{2}
dveg.

Moreover, we use another test function,c_{δ}(r)−ce_{δ} withce_{δ} = 1
V(M)

R

Mc_{δ}(r)dv

eg. α1

Z

M

(cδ(r)−ceδ)^{2}dv

eg 6 Z

Ω

h∇(cδ(r)−ceδ),∇(cδ(r)−ceδ)idvg

+b Z

M

D

∇e(cδ(r)−ceδ),∇e(cδ(r)−ceδ)E dveg

6 Z

Ω

h∇cδ(r),∇cδ(r)idvg+b Z

M

D

∇ce δ(r),∇ce δ(r)E dveg

6 δ^{2}
Z

Ω

s^{2}_{δ}(r)h∇r,∇ridvg+bδ^{2}
Z

M

s^{2}_{δ}(r)D

∇r,e ∇re E dveg

6 δ^{2}
Z

Ω

kX^{>}k^{2}dvg+bδ^{2}
Z

M

kX^{>}^{e}k^{2}dv

eg,

where X^{>} =s_{δ}(r)∇r is the tangent component of X to Ω andX^{>}^{e} =s_{δ}(r)∇re is
the part ofX tangent toM. From this, we get

(23)

α1

Z

M

cδ(r)^{2}dv

eg6δ^{2}
Z

Ω

kX^{>}k^{2}dvg+bδ^{2}
Z

M

kX^{>}^{e}k^{2}dv

eg

+ α1

V(M) Z

M

cδ(r)dv

eg

^{2}
.

Hence summing (23) andδtimes (22), using the fact that c^{2}_{δ}+δs^{2}_{δ} = 1, gives
α1V(M)6δ

nV(Ω) +b(n−1)V(M) + α1

V(M) Z

M

cδ(r)dv

eg

2

, and so

α1V(M) 1− 1

V(M) Z

M

cδ(r)dv

eg

2! 6δ

nV(Ω) +b(n−1)V(M) . Moreover, from we have Lemma 3.2 (withf = 0), we have

1−

Z

M

cδ(r)dv

eg

V(M)

2

> 1

1 + Z

M

kH_{S}k^{2}dv

eg

δinf(tr (S))^{2}V(M)
which gives

α_{1}V(M)6δ

nV(Ω) +b(n−1)V(M)

δ+

Z

M

kHSk^{2}dv

eg

V(M) inf (tr (S))^{2}

and finally the desired estimate

α16

nV(Ω)

V(M)+b(n−1)

δ+

Z

M

kH_{S}k^{2}dv

eg

V(M) inf (tr (S)^{2})

.

This concludes the proof.

5. A remark about the caseδ >0

The aim of this section is to compare the estimates in both case δ > 0 and
δ60. Indeed, in the estimates of Theorems 1.2, the radiusRof a ball containing
Ω appears when δ60 but not for the estimates forδ >0. It turns out that when
δ >0, we can bound the radiusRfrom above in terms ofH_{T} and trT and so obtain
upper bounds comparable to those obtained forδ60. First, we have the following
Proposition 5.1. Let ( ¯M^{n+p},¯g,µ¯_{f})be a weighted Riemannian manifold with sec-
tional curvaturesectM¯ 6δ, with δ >0, andµ¯f =e^{−f}dv¯g. Let(Ω, g)be a compact
Riemannian manifold with non-empty boundary M isometrically immersed into
( ¯M^{n+p},g)¯ by X. We endow Ω with the weighted measure µf = e^{−f}dvg. Let T
be a symmetric, divergence-free and positive definite (1,1)-tensor on Ω. If Ω is
contained a ball of radius R, then

s^{2}_{δ}(R)

kH_{T} −T∇fk^{2}_{∞}
infΩ(tr (T))^{2} +δ

>1.

Proof. From the second part of Lemma 2.1 applied on Ω, we have Z

Ω

cδ(r)tr (T)µf 6− Z

Ω

hX, HT −T∇fiµf. We recall thatX =sδ(r)∇r, which gives

Z

Ω

cδ(r)tr (T)µf 6 Z

Ω

kHT −T∇fksδ(r)µf.

We are in the case where δ > 0, so c_{δ} and s_{δ} are respectively decreasing and
increasing on [0, ^{π}

2√

δ]. Moreover, Ω is contained in a ball of radius R < ^{π}

4√
δ which
implies that Ω is contained in a ball of radius ^{π}

2√

δ centered atp. Hence,r < Rand socd(r)>cδ(R) andsδ(r)6sδ(R) on Ω. Thus, we get

cδ(R) inf

Ω(tr (T))6sδ(R) sup

Ω

kHT−T∇fk.

We deduce easily from this and the fact thatc^{2}_{δ}+δs^{2}_{δ} = 1 that
s^{2}_{δ}(R)

kHT −T∇fk^{2}_{∞}
inf

Ω(tr (T))^{2} +δ

>1,

which concludes the proof of the proposition.

Now, using the above proposition together with the estimate of Theorem 1.2 in the caseδ >0, we get the following estimate

σ1 6

kHT−T∇fk^{2}_{∞}
inf

Ω(tr (T))^{2} +δ

δ+

Z

M

kHS−S(∇f)k^{2}µe_{f}
Vf(M) inf (tr (S)^{2})

Z

Ω

tr (T)µ_{f}
Vf(M) s^{2}_{δ}(R)

6 sup

Ω

(tr (T))

kHT−T∇fk^{2}_{∞}
inf

Ω(tr (T))^{2} +δ

δ+

Z

M

kH_{S}−S(∇f)k^{2}µe_{f}
Vf(M) inf (tr (S)^{2})

Vf(Ω)
Vf(M)s^{2}_{δ}(R)
which is comparable to the estimate for the caseδ60:

σ16sup

Ω

δtr (T) + sup

Ω

kHT −T(∇f)k tr (T)

kHT −T(∇f)k

×

δ+sup_{M}kHS−S(∇f)k^{2}
inf_{M}(tr (S))^{2}

Vf(Ω)
V_{f}(M)s^{2}_{δ}(R).

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Institute of Mathematics and Computer Science University of S˜ao Paulo, S˜ao Carlos, Brazil Email address:manfio@icmc.usp.br

Laboratoire d’Analyse et de Math´ematiques Appliqu´ees, UPEM-UPEC, CNRS, F-77454 Marne-la-Vall´ee, France Email address:julien.roth@u-pem.fr

School of Mathematice and Computer Science,

Indian Institute Of Technology Goa, Ponda, 403401, India Email address:abhitosh@iitgoa.ac.in