1 Introduction
Let
$(M^n,g)$
be an n-dimensional compact, connected, oriented manifold without boundary, isometrically immersed into the
$(n+1)$
-dimensional Euclidean space
$\mathbb {R}^{n+1}$
. The spectrum of the Laplacian of
$(M,g)$
is an increasing sequence of real numbers
$$ \begin{align*} 0=\lambda_0(\Delta)<\lambda_1(\Delta)\leqslant\lambda_2(\Delta)\leqslant\cdots\leqslant \lambda_k(\Delta)\leqslant\cdots\longrightarrow +\infty. \end{align*} $$
The eigenvalue
$0$
(corresponding to constant functions) is simple and
$\lambda _1(\Delta )$
is the first positive eigenvalue. In [Reference Reilly16], Reilly proved the following well-known upper bound for
$\lambda _1(\Delta )$
:
$$ \begin{align} \lambda_1(\Delta)\leqslant\frac{n}{\mathrm{V}(M)}\int_MH^2\,dv_g, \end{align} $$
where H is the mean curvature of the immersion. He also proved an analogous inequality involving the higher order mean curvatures: for
$r\in \{1,\ldots ,n\}$
,
$$ \begin{align} \lambda_1(\Delta)\bigg(\int_MH_{r-1}\,dv_g\bigg)^2\leqslant\mathrm{V}(M)\int_MH_r^2\,dv_g, \end{align} $$
where
$H_r$
is the rth mean curvature, defined by the rth symmetric polynomial of the principal curvatures.
Inequalities (1.1) and (1.2) have been generalised in many ways for submanifolds of any codimension of Euclidean spaces and spheres [Reference Grosjean9, Reference Reilly16], submanifolds of hyperbolic spaces [Reference El Soufi and Ilias7, Reference Grosjean9, Reference Heintze11], other differential operators of divergence-type [Reference Alias and Malacarné1, Reference Roth17] and Paneitz-like operators [Reference Roth18], as well as for different types of Steklov problems. In particular, Ilias and Makhoul [Reference Ilias and Makhoul12] proved the following upper bound for the first eigenvalue
$\sigma _1$
of the Steklov problem:
$$ \begin{align*}\sigma_1V(\partial M)^2\leqslant nV(M)\int_{\partial M}\|H\|^2\,dv_g,\end{align*} $$
where
$(M^n,g)$
is a compact submanifold of
$\mathbb {R}^N$
with boundary
$\partial M$
and H denotes the mean curvature of
$\partial M$
. They also proved analogous inequalities involving higher order mean curvatures as in (1.2). Recently, Manfio and the authors have extended this inequality for submanifolds of any Riemannian manifold of bounded sectional curvature in [Reference Manfio, Roth and Upadhyay15].
Let us consider
$(M^n,g)$
, a compact Riemannian manifold with a possibly nonempty boundary
$\partial M$
. For
$p\in (1,+\infty )$
, we consider the so-called p-Laplacian defined by
$$ \begin{align*}\Delta_pu=-\mathrm{div}(\|\nabla u\|^{p-2}\nabla u)\end{align*} $$
for any
$\mathcal {C}^2$
function u. For
$p=2$
,
$\Delta _2$
is just the Laplace–Beltrami operator of
$(M^n,g)$
. This operator
$\Delta _p$
and especially its spectrum have been intensively studied, mainly for Euclidean domains with Dirichlet or Neumann boundary conditions (see for instance [Reference Lê13] and references therein) and also on Riemannian manifolds [Reference Lima, Montenegro and Santos14]. Later, Du and Mao [Reference Du and Mao6] gave analogues of the Reilly inequalities (1.1) and (1.2) for the p-Laplacian on submanifolds of Euclidean spaces and spheres and it was extended by Chen and Wei [Reference Chen and Wei5] for submanifolds of hyperbolic space. Very recently, Chen [Reference Chen3] and Chen and Gui [Reference Chen and Gui4] have obtained upper bounds for submanifolds of manifolds with curvature bounded from above, generalising to the p-Laplacian the result of Heintze for the Laplacian.
In the present paper, we will consider the Steklov problem associated with the p-Laplacian on submanifolds with boundary of the Euclidean space. That is, we consider the p-Steklov problem which is the following boundary value problem:
$$\begin{align*} \begin{cases} \Delta_pu=0&\text{in}\ M,\\ \|\nabla u\|^{p-2}\dfrac{\partial u}{\partial\nu}=\sigma |u|^{p-2}u&\text{on}\ \partial M, \end{cases} \end{align*}$$
where
${\partial u}/{\partial \nu }$
is the derivative of the function u with respect to the outward unit normal
$\nu $
to the boundary
$\partial M$
. Note that for
$p=2$
, (
$S_p$
) is the usual Steklov problem. (See, for example, [Reference Girouard and Polterovich8] for an overview of results about the spectral geometry of the Steklov problem.) It has been observed that very little is known about the spectrum of the p-Steklov problem. If M is a domain of
$\mathbb {R}^N$
, there exists a sequence of positive eigenvalues
$\sigma _{0,p}=0<\sigma _{1,p}\leqslant \sigma _{2,p}\leqslant \cdots \leqslant \sigma _{k.p}\leqslant \cdots $
in the variational spectrum obtained by the Ljusternik–Schnirelmann theory (see [Reference Lê13, Reference Torné20] and also [Reference Browder, Chern and Smale2] for details of the Ljusternik–Schnirelmann principle). Note that, as mentioned in [Reference Lima, Montenegro and Santos14, Remark 1.1], the arguments used in [Reference Lê13] can be extended to domains on Riemannian manifolds and there exists a nondecreasing sequence of variational eigenvalues obtained by the Ljusternik–Schnirelman principle. Moreover, the eigenvalue
$0$
is simple with constant eigenfunctions and isolated, that is, there is no eigenvalue between
$0$
and
$\lambda _1$
. The first positive eigenvalue of the p-Steklov problem is
$\sigma _{1,p}$
and it has the variational characterisation
$$ \begin{align} \sigma_{1,p}=\inf\bigg\{ \frac{\int_{M}\|\nabla u\|^p\,dv_g}{\int_{\partial M}|u|^p\,dv_h} \ \bigg|\ u\in W^{1,p}(M)\setminus\{0\},\ \int_{\partial M}|u|^{p-2}u\,dv_h=0\bigg\}, \end{align} $$
where
$\nabla $
is the gradient on M, and
$dv_g$
and
$dv_h$
are the Riemannian volume forms respectively associated with the metric g on M and the induced metric h on
$\partial M$
. All the other eigenvalues
$\sigma _{k,p}$
of this sequence also have a variational characterisation but we do not know if all the spectrum is contained in this sequence.
Recently, Verma obtained upper bounds for the first eigenvalue
$\sigma _{1,p}$
of the p-Steklov problem (
$S_p$
) for Euclidean domains [Reference Verma21]. She proved that for a bounded domain
$\Omega $
with smooth boundary,
$\sigma _{1,p}\leqslant {1}/{R^{p-1}}$
if
$1<p<2$
and
$\sigma _{1,p}\leqslant {n^{p-2}}/{R^{p-1}}$
if
$p\geqslant 2$
, where
$R>0$
satisfies
$\mathrm {V}(\Omega )=\mathrm {V}(B(R))$
and
$B(R)$
is a ball of radius R. After that, in [Reference Roth19], the first author proved the following upper bounds of Reilly-type for
$\sigma _{1,p}$
for submanifolds with boundary of the Euclidean space:
$$ \begin{align*}\sigma_{1,p}\leqslant N^{{|2-p|}/{2}}n^{{p}/{2}}\bigg( \int_{\partial M}\|H\|^{{p}/{(p-1)}}\,dv_h\bigg)^{p-1}\frac{V(M)}{V(\partial M)^p}\end{align*} $$
and, more generally,
$$ \begin{align*}\sigma_{1,p}\bigg(\int_{\partial M}\mathrm{tr}\,(T)\,dv_h\bigg)^p\leqslant N^{{|2-p|}/{2}}n^{{p}/{2}}\bigg( \int_{\partial M}\|H_T\|^{{p}/{(p-1)}}\,dv_h\bigg)^{p-1}V(M),\end{align*} $$
where T is a symmetric and divergence-free
$(1,1)$
-tensor on
$\partial M$
. The aim of the present paper is to prove an inequality for submanifolds with boundary of Riemannian manifolds of sectional curvature bounded from above by a nonnegative constant. We prove the following result.
Theorem 1.1. Let
$\delta \geqslant 0$
,
$p>1$
be real numbers and
$(\bar {M}^N,\bar {g})$
an N-dimensional Riemannian manifold of sectional curvature bounded from above by
$\delta $
. Let
$(M^n,g)$
be a compact n-dimensional Riemannian manifold with nonempty boundary
$\partial M$
isometrically immersed into
$(\bar {M},\bar {g})$
and let S be a symmetric, positive definite and divergence-free
$(1,1)$
-tensor on
$\partial M$
.
-
(1) If
$\delta =0$
, then
$$ \begin{align*}\sigma_{1,p}\bigg(\int_{\partial M}\mathrm{tr}\,(S)\,dv_h\bigg)^p\leqslant N^{{|p-2|}/{2}}n^{{p}/{2}}V(M)\bigg( \int_{\partial M}\|H_S\|^{{p}/{(p-1)}}\,dv_h\bigg)^{p-1}.\end{align*} $$
-
(2) If
$\delta>0$
and M is contained in a ball of radius
$R\leqslant {\pi }/{4\sqrt {\delta }}$
, then-
(a) for
$1<p<2$
, we have
$$ \begin{align*} \sigma_{1,p}\leqslant\delta^{({p}/{2})-1}(N+1)^{{(2-p)}/{2}}n^{{p}/{2}}\dfrac{V(M)}{V(\partial M)}\bigg(\delta+\frac{\int_{\partial M}\|H_S\|^2\,dv_g}{ \inf(\mathrm{tr}\,(S))^2V(\partial M)}\bigg);\end{align*} $$
-
(b) for
$p\geqslant 2$
, we have
$$ \begin{align*}\sigma_{1,p}\leqslant(N+1)^{({p-2)}/{2}}n^{{p}/{2}}\frac{V(M)}{V(\partial M)}\bigg(\delta+\frac{\int_{\partial M}\|H_S\|^2\,dv_g}{ \inf(\mathrm{tr}\,(S))^2V(\partial M)}\bigg)^{{p}/{2}}.\end{align*} $$
-
2 Preliminaries
Let
$(\bar {M}^{N},\bar {g})$
be an N-dimensional Riemannian manifold with sectional curvature
$K_{\bar {M}}\leqslant \delta $
. For q a fixed point in
$\bar {M}$
, we denote by
$r(x)$
the geodesic distance between x and q, and we define the vector field Z by
$Z(x):=s_{\delta }(r(x))(\bar {\nabla } r)(x)$
, where
$s_{\delta }$
is the function defined by
$$ \begin{align*} s_{\delta}(r)= \begin{cases} \dfrac{1}{\sqrt{\delta}} \sin(\sqrt{\delta}r) &\; \text{if}\ \delta>0,\\ r &\; \text{if}\ \delta=0, \\ \dfrac{1}{\sqrt{|\delta|}}\sinh(\sqrt{|\delta|}r) &\; \text{if}\ \delta<0.\ \ \end{cases}\end{align*} $$
We also define
$$ \begin{align*} c_{\delta}(r)=\begin{cases} \cos(\sqrt{\delta}r) &\; \text{if}\ \delta>0,\\ 1 &\; \text{if}\ \delta=0, \\ \cosh(\sqrt{|\delta|}r) &\; \text{if}\ \delta<0.\ \ \end{cases}\end{align*} $$
Hence,
$c_{\delta }^2+\delta s_{\delta }^2=1$
,
$s_{\delta }'=c_{\delta }$
and
$c_{\delta }'=-\delta s_{\delta }$
.
To prove Theorem 1.1, we recall some key lemmas. The first, in some sense, extends the Hsiung–Minkowki formulas to spaces of nonconstant curvature.
Lemma 2.1 (Grosjean, [Reference Grosjean10]).
Let
$(\Sigma ,g)$
be a compact submanifold of
$(\bar {M},\bar {g})$
and S be a symmetric, positive definite and divergence-free
$(1,1)$
-tensor on
$\Sigma $
. Then
-
(1)
$\displaystyle \sum _{i=1}^N\langle S\nabla Z_i,Z_i\rangle \leqslant \mathrm {tr}\,(S)-\delta \langle SZ^{\perp },Z^{\perp }\rangle ;$
-
(2)
$\mathrm {div}(SZ^{\top })\geqslant (c_{\delta }(r)\mathrm {tr}\,(S)+\langle Z,H_S\rangle ).$
If in addition,
$\Sigma $
has no boundary,
-
(3)
$\displaystyle \int _{\Sigma }c_{\delta }(r)\mathrm {tr}\,(S)\,dv_g\leqslant \displaystyle \int _{\Sigma }\|H_S\|s_{\delta }(r)\,dv_g;$
-
(4)
$\delta \displaystyle \int _{\Sigma } \langle SZ^{\top },Z^{\top }\rangle \,dv_g\geqslant \displaystyle \int _{\Sigma }(c_{\delta }^2(r)\mathrm {tr}\,(S)-\|H_S\|s_{\delta }(r)c_{\delta }(r))\,dv_g$
.
Here,
$H_S$
denotes
$\mathrm {tr}\,(B\circ S)$
and so is a normal vector field and
$Z^{\top }$
is the part of Z tangent to
$\Sigma $
. Note that if
$S=\mathrm {Id}\,$
, we recover the classical inequalities proved by Heintze [Reference Heintze11].
To prove the desired upper bounds, we will use the variational characterisation (1.3) of
$\sigma _{1,p}$
. For this, we need to use appropriate test functions. As usual, for eigenvalue upper bounds for submanifolds, the candidates for test functions are the coordinate functions and their analogues in nonconstant curvature
$Z_i=({s_{\delta }(r)}/{r})x_i$
,
$1\leqslant i \leqslant N$
, which are the coordinates of Z in a normal frame
$\{e_1,\ldots ,e_N\}$
. To be eligible to be test functions, we need to ‘centre’ these functions using the following lemma given by Chen in [Reference Chen3].
Lemma 2.2 [Reference Chen3].
Let
$p\in (1,+\infty )$
and assume that
$\Sigma $
is a submanifold of
$\bar {M}$
contained in a convex ball
$B\subset \bar {M}$
. Then, there exists
$q_0\in B$
such that for any
$i\in \{1,\ldots ,N\}$
,
$$ \begin{align*}\int_{\Sigma} \bigg|\dfrac{s_{\delta}(r)}{r}x_i\bigg|^{p-2}\dfrac{s_{\delta}(r)}{r}x_i\,dv_g=0,\end{align*} $$
where r is the distance function to
$q_0$
in
$\bar {M}$
.
For the case
$\delta>0$
, we need another test function
$c_{\delta }$
. To use it as a test function, we need to translate it appropriately. For this, we recall the following elementary lemma, also given by Chen in [Reference Chen3].
Lemma 2.3. Let
$\delta>0$
,
$p\in (1,+\infty )$
and assume that
$\Sigma $
is a submanifold of
$\bar {M}$
contained in a ball of centre
$q_0$
and radius
$\rho \leq {\pi }/{2\sqrt {\delta }}$
. Then, there exists a constant
$c\in [0,1]$
so that
$$ \begin{align*} \displaystyle\int_{\Sigma}\bigg|\dfrac{c_{\delta}(r)-c}{\sqrt{\delta}} \bigg|^{p-2}\dfrac{c_{\delta}(r)-c}{\sqrt{\delta}}\,dv_g=0, \end{align*} $$
where r is the distance function to
$q_0$
in
$\bar {M}$
.
Finally, we recall the following technical lemma proved by Manfio and the authors in [Reference Manfio, Roth and Upadhyay15] which will be useful at the end of the proof of Theorem 1.1.
Lemma 2.4 [Reference Manfio, Roth and Upadhyay15].
Let
$(\bar {M}^{N},\bar {g})$
be a Riemannian manifold with sectional curvature bounded from above by
$\delta $
,
$\delta>0$
. Let
$(\Sigma ,g)$
be a closed Riemannian manifold isometrically immersed into
$(\bar {M}^{N},\bar {g})$
and assume that
$\Sigma $
is contained in a geodesic ball of radius
$R\leq {\pi }/{2\sqrt {\delta }}$
. Let S be a symmetric, divergence-free and positive definite
$(1,1)$
-tensor on
$\Sigma $
. Then, we have
$$ \begin{align*}1-\Bigg(\frac {\int_{\Sigma}c_{\delta}(r)\,dv_g}{V(\Sigma)}\Bigg)^2\geqslant\frac{1}{1+\dfrac{\int_{\Sigma}\|H_S\|^2\,dv_g}{\delta \inf(\mathrm{tr}\,(S))^2V(\Sigma)}}.\end{align*} $$
3 Proof of Theorem 1.1
3.1 The case
$\delta =0$
To use the coordinate functions as test functions in the variational characterisation of
$\sigma _{1,p}$
, we need to place the coordinate centre at a good point. Therefore, we apply Lemma 2.2 to
$\Sigma =\partial M$
and we consider r as the distance to the point
$q_0$
given in Lemma 2.2. Thus, we are able to prove the following lemma.
Lemma 3.1. For any
$p\in (1,+\infty )$
,
$$ \begin{align*}\sigma_{1,p}\int_{\partial M}r^p\,dv_h\leqslant N^{{|p-2|}/{2}}n^{p/2}V(M).\end{align*} $$
Proof. From Lemma 2.2, we can consider the functions
$Z_i=({s_{\delta }(r)}/{r})x_i$
,
$1\leqslant i\leqslant N$
, as test functions in the variational characterisation (1.3) of
$\sigma _{1,p}$
. For
$\delta =0$
, we have
$Z_i=x_i$
. Taking the summation for i from
$1$
to N,
$$ \begin{align} \sigma_{1,p}\int_{\partial M}\displaystyle\sum_{i=1}^N|Z_i|^p\,dv_h \leqslant \int_{M}\displaystyle\sum_{i=1}^N\|\nabla Z_i\|^p\,dv_g. \end{align} $$
We will discuss the cases
$p\geqslant 2$
and
$1<p<2$
, separately.
Case 1:
$1<p<2$
. Since
$p<2$
,
$$ \begin{align} r^p=( r^2)^{{p}/{2}}=\bigg( \displaystyle\sum_{i=1}^N Z_i^2\bigg)^{{p/2}}\leqslant\displaystyle\sum_{i=1}^N |Z_i|^p. \end{align} $$
However, by the Hölder inequality (for vectors),
$$ \begin{align} \displaystyle\sum_{i=1}^N \|\nabla Z_i\|^p\leqslant N^{{(2-p)}/{p}}\bigg( \displaystyle\sum_{i=1}^N \|\nabla Z_i\|^2\bigg)^{{p}/{2}}, \end{align} $$
which gives with Lemma 2.1(1) and
$\delta =0$
,
$$ \begin{align} \displaystyle\sum_{i=1}^N \|\nabla Z_i\|^p\leqslant N^{{(2-p)}/{p}}n^{{p}/{2}}. \end{align} $$
$$ \begin{align} \sigma_{1,p}\int_{\partial M}r^p\,dv_h\leqslant\sigma_{1,p}\int_{\partial M}\displaystyle\sum_{i=1}^N|Z_i|^p\,dv_h \leqslant \int_{M}\displaystyle\sum_{i=1}^N\|\nabla Z_i\|^p\,dv_g \leqslant N^{{(2-p)}/{2}}n^{p/2}V(M). \end{align} $$
Case 2:
$p\geqslant 2$
. By the Hölder inequality,
$$ \begin{align*} r^2=\displaystyle\sum_{i=1}^N| Z_i|^2\leqslant N^{{(p-2)}/{p}}\bigg( \displaystyle\sum_{i=1}^N| Z_i|^p\bigg)^{{2}/{p}}, \end{align*} $$
which gives
$$ \begin{align} r^p= N^{{(p-2)}/{2}}\bigg( \displaystyle\sum_{i=1}^N| Z_i|^p\bigg). \end{align} $$
Moreover, since
$p\geqslant 2$
,
$$ \begin{align} \displaystyle\sum_{i=1}^N\| \nabla Z_i\|^p\leqslant \bigg(\displaystyle\sum_{i=1}^N\|\nabla Z_i\|^2\bigg) ^{{p}/{2}}. \end{align} $$
Finally, from (3.1), using (3.6), (3.7) and Lemma 2.1(1),
$$ \begin{align*} \sigma_{1,p}\int_{\partial M}r^p\,dv_h&\leqslant \sigma_{1,p}N^{{(p-2)}/{2}}\sigma_{1,p}\int_{\partial M}\displaystyle\sum_{i=1}^N|Z_i|^p\,dv_h\\ &\leqslant N^{{(p-2)}/{2}}\int_{M}\displaystyle\sum_{i=1}^N\|\nabla Z_i\|^p\,dv_g\\ &\leqslant N^{{(p-2)}/{2}}\int_{M}\bigg(\displaystyle\sum_{i=1}^N\|\nabla Z_i\|^2\bigg) ^{{p}/{2}}\,dv_g\\ &\leqslant N^{({p-2)}/{2}}n^{p/2}V(M). \\[-2.5pc] \end{align*} $$
Since
$\delta =0$
, we have
$c_{\delta }\equiv 1$
and Lemma 2.1(3) reduces to
$$ \begin{align*}\int_{\partial M}\mathrm{tr}\,(S)\,dv_h\leqslant\int_{\partial M}s_{\delta}(r)\|H_S\|\,dv_h.\end{align*} $$
Thus,
$$ \begin{align*} \sigma_{1,p}\bigg(\int_{\partial M}\mathrm{tr}\,(S)\,dv_h\bigg)^p&\leqslant \sigma_{1,p}\bigg(\int_{\partial M}s_{\delta}(r)\|H_S\|\,dv_h\bigg)^p\\ &\leqslant \sigma_{1,p}\bigg( \int_{\partial M}s_{\delta}^p(r)\,dv_h\bigg)\bigg( \int_{\partial M}\|H_S\|^{{p}/{(p-1)}}\,dv_h\bigg)^{p-1}\\ &\leqslant N^{{|p-2|}/{2}}n^{p/2}V(M)\bigg( \int_{\partial M}\|H_S\|^{{p}/{(p-1)}}\,dv_h\bigg)^{p-1}, \end{align*} $$
where we have used first the Hölder inequality and then Lemma 3.1 since
$s_\delta (r) = r$
when
$\delta = 0$
.
3.2 The case
$\delta>0$
In the case
$\delta>0$
, in addition to the
$Z_i$
terms, we need another test function. For this, from the assumption that M is contained in a ball of radius
$R\leq {\pi }/{4\sqrt {\delta }}$
and since the point
$q_0$
in Lemma 2.2 belongs to this ball, we can conclude that M is contained in a ball of centre
$q_0$
and radius smaller than or equal to
${\pi }/{2\sqrt {\delta }}$
. Therefore, we can apply Lemma 2.3 to get a constant
$c\in [0,1]$
so that
$$ \begin{align*} \displaystyle\int_{\Sigma}\bigg|\dfrac{c_{\delta}(r)-c}{\sqrt{\delta}} \bigg|^{p-2}\dfrac{c_{\delta}(r)-c}{\sqrt{\delta}}\,dv_g=0. \end{align*} $$
The function
$C={(c_{\delta }(r)-c)}/{\sqrt {\delta }}$
can be used as a test function. From the variational characterisation (1.3) of
$\sigma _{1,p}$
using C and the
$Z_i$
terms as test functions, we get
$$ \begin{align} \sigma_{1,p}\displaystyle\int_{\partial M}\bigg(|C|^p+\sum_{i=1}^N|Z_i|^p\bigg)\,dv_h\leqslant\displaystyle\int_{M}\bigg(\|\nabla C\|^p+\displaystyle\sum_{i=1}^N\|\nabla Z_i\|^p\bigg)\,dv_g. \end{align} $$
Moreover,
$$ \begin{align} C^2+\displaystyle\sum_{i=1}^NZ_i^2&=\bigg( \dfrac{c_{\delta}(r)-c}{\sqrt{\delta}}\bigg)^2+\displaystyle\sum_{i=1}^N\bigg(\frac{s_{\delta}(r)}{r}x_i\bigg)^2\nonumber\\[-1pt] &=s_{\delta}^2(r)+\frac{c_{\delta}^2(r)+c^2-2cc_{\delta}(r)}{\delta}\nonumber\\[-1pt] &=\frac{1+c^2-2cc_{\delta}(r)}{\delta}, \end{align} $$
where we have used
$c_{\delta }^2+\delta s_{\delta }^2=1$
. However, we also have
$$ \begin{align*} \nabla C=\nabla\bigg( \dfrac{c_{\delta}(r)-c}{\sqrt{\delta}}\bigg)=-\sqrt{\delta}s_{\delta}(r)\nabla r=\sqrt{\delta}Z^{\perp}, \end{align*} $$
so that
$$ \begin{align} \|\nabla C\|^2+\displaystyle\sum_{i=1}^N\|\nabla Z_i\|^2=\delta\|Z^{\perp}\|^2+\displaystyle\sum_{i=1}^N\|\nabla Z_i\|^2 \leqslant\delta\|Z^{\perp}\|^2+(n-\delta\|Z^{\perp}\|^2) =n, \end{align} $$
where we have used Lemma 2.1(1). Note here that
$Z^{\top }$
is the part of Z tangent to M. We now consider the cases
$1<p<2$
and
$p\geqslant 2$
separately.
Case 1:
$1<p<2$
. Since
$p<2$
,
$$ \begin{align} |C|^p+\displaystyle\sum_{i=1}^N |Z_i|^p=\frac{1}{\delta^{{p}/{2}}}\bigg( |{\cos}(\sqrt{\delta}\,r)-c|^p+\displaystyle\sum_{i=1}^N \bigg{|}{\sin}(\sqrt{\delta}\,)\frac{x_i}{r}\bigg|^p\bigg). \end{align} $$
Since
$|{\sin} (\sqrt {\delta }\,){x_i}/{r}|\leqslant 1$
,
$|{\cos} (\sqrt {\delta }\,r)-c|<1$
and
$1<p<2$
,
$$ \begin{align*} \bigg{|}{\sin}(\sqrt{\delta}\,)\frac{x_i}{r}\bigg|^p\geqslant \bigg{|}{\sin}(\sqrt{\delta}\,)\frac{x_i}{r}\bigg|^2\quad \text{and}\quad |{\cos}(\sqrt{\delta}\,r)-c|^p\geqslant |{\cos}(\sqrt{\delta}\,r)-c|^2, \end{align*} $$
which after substituting into (3.11) gives
$$ \begin{align} |C|^p+\displaystyle\sum_{i=1}^N |Z_i|^p&\geqslant \frac{1}{\delta^{{p}/{2}}}\bigg( |{\cos}(\sqrt{\delta}\,r)-c|^2+\displaystyle\sum_{i=1}^N \bigg|{\sin}(\sqrt{\delta}\,)\frac{x_i}{r}\bigg|^2\bigg)\nonumber\\[-1pt] &= \frac{1}{\delta^{({p}/{2})-1}}\bigg( C^2+\displaystyle\sum_{i=1}^N Z_i^2\bigg)\nonumber\\[-1pt] &= \frac{1}{\delta^{{p}/{2}}}(1+c^2-2cc_{\delta}(r)), \end{align} $$
where we have used (3.9) for the last line. However, by the Hölder inequality,
$$ \begin{align} \|\nabla C\|^p+\displaystyle\sum_{i=1}^N \|\nabla Z_i\|^p &\leqslant(N+1)^{{(2-p)}{2}}\bigg( \|\nabla C\|^2+\displaystyle\sum_{i=1}^N \|Z_i\|^2\bigg)^{{p}/{2}} \nonumber\\[4pt] &\leqslant(N+1)^{{(2-p)}{2}}n^{{p}/{2}} \end{align} $$
by using (3.10). From (3.8) together with (3.12) and (3.13),
$$ \begin{align} \sigma_{1,p}\displaystyle\int_{\partial M}(1+c^2-2cc_{\delta}(r))\,dv_h \leqslant \delta^{{p}/{2}}(N+1)^{{(2-p)}/{2}}n^{{p}/{2}}V(M). \end{align} $$
Moreover,
$$ \begin{align} \int_{\partial M}(1+c^2-2cc_{\delta}(r))\,dv_h&=V(\partial M)\bigg(1+c^2-2c\frac{\int_{\partial M}c_{\delta}(r)\,dv_h}{V(\partial M)}\bigg)\nonumber\\[4pt] &=V(\partial M)\bigg(1+\bigg(c-\frac{\int_{\partial M}c_{\delta}(r)\,dv_h}{V(\partial M)}\bigg)^2-\bigg(\frac{\int_{\partial M}c_{\delta}(r)\,dv_h}{V(\partial M)}\bigg)^2\bigg)\nonumber\\[4pt] &\geqslant V(\partial M)\bigg(1-\bigg(\frac{\int_{\partial M}c_{\delta}(r)\,dv_h}{V(\partial M)}\bigg)^2\bigg). \end{align} $$
Substituting this into (3.14) yields
$$ \begin{align} \sigma_{1,p}V(\partial M)\bigg(1-\bigg(\frac{\int_{\partial M}c_{\delta}(r)\,dv_h}{V(\partial M)}\bigg)^2\bigg)\,dv_h \leqslant \delta^{{p}/{2}}(N+1)^{{(2-p)}/{2}}n^{{p}/{2}}V(M). \end{align} $$
Finally, by Lemma 2.4,
$$ \begin{align*} \sigma_{1,p} \leqslant \delta^{{p}/{2}-1}(N+1)^{{(2-p)}/{2}}n^{{p}/{2}}\frac{V(M)}{V(\partial M)}\bigg(\delta+\dfrac{\int_{\partial M}\|H_S\|^2\,dv_g}{ \inf(\mathrm{tr}\,(S))^2V(\partial M)}\bigg). \end{align*} $$
Case 2:
$p\geqslant 2$
. Since
$p\geqslant 2$
,
$$ \begin{align} \|\nabla C\|^p+\displaystyle\sum_{i=1}^N \|Z_i\|^p \leqslant \bigg(\|\nabla C\|^2+\displaystyle\sum_{i=1}^N \|Z_i\|^2\bigg)^{{p}/{2}} \leqslant n^{{p}/{2}}, \end{align} $$
where we have used (3.10). However, by the Hölder inequality,
$$ \begin{align} \bigg(C^2+\displaystyle\sum_{i=1}^N Z_i^2\bigg)^{{p}/{2}} \leqslant (N+1)^{{(p-2)}/{2}}\bigg(|C|^p+\displaystyle\sum_{i=1}^N |Z_i|^p\bigg). \end{align} $$
Thus, using successively (3.9), (3.18), (3.8) and (3.17),
$$ \begin{align} \sigma_{1,p/2}\displaystyle\int_{\partial M}\bigg( \frac{1+c^2-2cc_{\delta}(r)}{\delta}\bigg)^{{p}{2}}\,dv_h &=\sigma_{1,p}\displaystyle\int_{\partial M}\bigg( C^2+\displaystyle\sum_{i=1}^N Z_i^2\bigg)^{{p}/{2}}\,dv_h\nonumber\\ &\leqslant \sigma_{1,p}(N+1)^{{(p-2)}/{2}}\displaystyle\int_{\partial M}\bigg( |C|^p+\displaystyle\sum_{i=1}^N |Z_i|^p\bigg)\,dv_h\nonumber\\ &\leqslant (N+1)^{{(p-2)}/{2}}\displaystyle\int_{M}\bigg( \|\nabla C\|^p+\displaystyle\sum_{i=1}^N \|\nabla Z_i\|^p\bigg)\,dv_h\nonumber\\ &\leqslant (N+1)^{{(p-2)}/{2}}n^{{p}/{2}}V(M). \end{align} $$
In addition, from the Hölder inequality (for integrals),
$$ \begin{align} \displaystyle\int_{\partial M}\bigg( \frac{1+c^2-2cc_{\delta}(r)}{\delta}\bigg)^{{p}/{2}}\,dv_h \geqslant V(\partial M)^{{(2-p)}/{2}}\bigg(\displaystyle\int_{\partial M}\frac{1+c^2-2cc_{\delta}(r)}{\delta}\,dv_h\bigg)^{{p}/{2}}. \end{align} $$
Hence, we deduce from (3.19) with (3.15) and (3.20),
$$ \begin{align} \sigma_{1,p}\dfrac{V(\partial M)}{\delta^{{p}/{2}}}\bigg[ 1-\bigg(\frac{\int_{\partial M}c_{\delta}(r)\,dv_h}{V(\partial M)}\bigg)^2\bigg]^{{p}/{2}} &\leqslant \sigma_{1,p}V(\partial M)^{{(2-p)}/{2}}\bigg(\displaystyle\int_{\partial M}\frac{1+c^2-2cc_{\delta}(r)}{\delta}\,dv_h\bigg)^{{p}/{2}}\nonumber\\ &\leqslant (N+1)^{{(p-2)}/{2}}n^{{p}/{2}}V(M). \end{align} $$
Finally, we use Lemma 2.4 to conclude that
$$ \begin{align*} \sigma_{1,p}\leqslant (N+1)^{{(p-2)}/{2}}n^{{p}/{2}}\dfrac{V(M)}{V(\partial M)}\bigg(\delta+\dfrac{\int_{\partial M}\|H_S\|^2\,dv_g}{ \displaystyle\inf_{\partial M}(\mathrm{tr}\,(S))^2V(\partial M)}\bigg)^{{p}/{2}}. \end{align*} $$
This completes the proof of Theorem 1.1.
4 New results for the p-Laplacian when
$\delta>0$
We finish by giving similar results for the first eigenvalue of the p-Laplacian for closed submanifolds when
$\delta>0$
. We will not give all the details of the proof since it is similar to the proof of Theorem 1.1. The difference is that the variational characterisation of
$\lambda _{1,p}$
is
$$ \begin{align*} \lambda_1=\inf\bigg\{ \dfrac{\int_{M}\|\nabla u\|^p\,dv_g}{\int_{M}|u|^p\,dv_g}\ \bigg|\ u\in W^{1,p}(M)\setminus\{0\},\ \int_{M}|u|^{p-2}u\,dv_g=0\bigg\}. \end{align*} $$
In this case, M has no boundary and so we can apply Lemmas 2.2 and 2.3 with
$\Sigma =M$
to use the functions C and
$Z_i$
,
$1\leqslant i\leqslant N$
, as test functions. By completely similar computations, we obtain the analogue of (3.16) if
$1<p<2$
, that is,
$$ \begin{align*}\lambda_{1,p}V(M)\bigg(1-\bigg(\frac{\int_{M}c_{\delta}(r)\,dv_h}{V(M)}\bigg)^2\bigg)\,dv_h\leqslant\delta^{{p}/{2}}(N+1)^{{(2-p)}/{2}}n^{{p}/{2}}V(M),\end{align*} $$
and of (3.21) if
$p\geq 2$
, that is,
$$ \begin{align*}\lambda_{1,p}\dfrac{V(M)}{\delta^{{p}/{2}}}\bigg[ 1-\bigg(\frac{\int_{M}c_{\delta}(r)\,dv_h}{V(M)}\bigg)^2\bigg]^{{p}/{2}}\leqslant(N+1)^{{(p-2)}/{2}}n^{{p}/{2}}V(M).\end{align*} $$
Finally, applying Lemma 2.4 to M, we deduce the following result.
Theorem 4.1. Let
$\delta>0$
,
$p\in (1,+\infty )$
and
$(\bar {M}^N,\bar {g})$
be a Riemannian manifold of sectional curvature bounded from above by
$\delta $
. Let
$(M^n,g)$
be a closed Riemannian manifold isometrically immersed into
$(\bar {M},\bar {g})$
and S a symmetric, positive definite and divergence-free
$(1,1)$
-tensor on
$\partial M$
. We denote by
$\lambda _{1,p}$
the first eigenvalue of the p-Laplacian on M. Suppose M is contained in a ball of radius
$R\leqslant {\pi }/{4\sqrt {\delta }}$
.
-
(1) If
$1<p<2$
,
$$ \begin{align*}\lambda_{1,p}\leqslant\delta^{({p}/{2})-1}(N+1)^{{(2-p)}/{2}}n^{{p}/{2}}\bigg(\delta+\dfrac{\int_{M}\|H_S\|^2\,dv_g}{ \displaystyle\inf_M(\mathrm{tr}\,(S))^2V(M)}\bigg).\end{align*} $$
-
(2) If
$p\geqslant 2$
,
$$ \begin{align*}\lambda_{1,p}\leqslant(N+1)^{{(p-2)}/{2}}n^{{p}/{2}}\bigg(\delta+\dfrac{\int_{M}\|H_S\|^2\,dv_g}{ \displaystyle\inf_M(\mathrm{tr}\,(S))^2V(M)}\bigg)^{{p}/{2}}.\end{align*} $$
If
$S=\mathrm {Id}$
, we recover the result of Chen [Reference Chen3].
