1 Erratum to: Geom Dedicata (2014) 169:397–410 DOI 10.1007/s10711-013-9863-0

In our paper “Sharp upper bounds for the first eigenvalue” [1], we proved the following theorem.

Theorem 1

Let \((\overline{M}, ds^2)\) be a non compact rank-1 symmetric space and \(M\) be a closed hypersurface in \(\overline{M}\) which encloses the bounded region \(\varOmega \). Then

$$\begin{aligned} \lambda _1(M) \le \lambda _1(S(R))\left( \frac{Vol(M)}{Vol(S(R))}\right) + \frac{1}{\sinh ^2R\,Vol(S(R))}\int _M \parallel \nabla ^M\sinh \,r \parallel ^2 \end{aligned}$$

where \(R>0\) is such that \(Vol(\varOmega ) = Vol(B(R))\); here \(B(R)\) and \(S(R)\) are the geodesic ball and geodesic sphere respectively of radius \(R\).

Further, the equality holds if and only if \(M\) is a geodesic sphere of radius \(R\).

The proof of this theorem uses the following inequality of [1]: For a closed hypersurface \(M\) in the noncompact rank-1 symmetric space \(\overline{M}\),

$$\begin{aligned} \lambda _1(M)\int _Mf^2 dm \le \int _M \parallel \nabla ^Mf \parallel ^2dm + \int _M f^2\left( \lambda _1(S(r))- \sum _{i=1}^{kn}\left( \frac{\partial f_i}{\partial \eta }\right) ^2\right) dm \end{aligned}$$

where \(f=\sinh r\).

Substituting the value of \(\lambda _1(S(r)) \) and applying Lemmas 1 and 2 of [1], we get

$$\begin{aligned} \lambda _1(M)Vol(S(R))\sinh ^2R&\le (kn-1)Vol(M) -(k-1)\tanh ^2R\,Vol(S(R))\nonumber \\&\quad + \int _M \parallel \nabla ^M\sinh \,r \parallel ^2dm \end{aligned}$$
(0.1)

when \( k = 1\), that is for \(\overline{M} = \mathbb {H}^n\), we get the required inequality stated in Theorem 1.

In a later inspection, we observed that when \( k>1\), we can not assert the validity of the inequality

$$\begin{aligned} \lambda _1(M)Vol(S(R))\sinh ^2R&\le \left( (kn-1) -(k-1)\tanh ^2R)\right) Vol(M)\\&\quad + \int _M \parallel \nabla ^M\sinh \,r \parallel ^2dm \end{aligned}$$

from Eq. (0.1). This was used to complete the proof Theorem 1.

As the above inequality does not hold in general, the Theorem 1 stated as above needs correction. The correct statement and proof of the theorem are as follows.

Theorem 2

Let \((\overline{M}, ds^2)\) be a non-compact rank-1 symmetric space with \(\text {dim}\,\overline{M} = kn \) where \(k = \text {dim}_{\mathbb {R}}\mathbb {K}; \, \mathbb {K} = \mathbb {R}, \mathbb {C}, \mathbb {H}\) or \(\mathbb {C}a\). Let \(M\) be a closed hypersurface in \(\overline{M}\) which encloses the bounded region \(\varOmega \). Then for \(k=1\), we have

$$\begin{aligned} \frac{\lambda _1(M)}{\lambda _1(S(R))}&\le \frac{Vol(M)}{Vol(S(R))} + \frac{1}{(n-1)Vol(S(R))}\int _M \parallel \nabla ^M\sinh \,r \parallel ^2 \end{aligned}$$

and for \(k>1\), we have

$$\begin{aligned} \lambda _1(M)&\le \lambda _1(S(R))\left( \frac{Vol(M)}{Vol(S(R))}\right) + \frac{k-1}{\cosh ^2R}\left( \frac{Vol(M)}{Vol(S(R))}\right) \\&\quad +\, \ \frac{1}{\sinh ^2R\,Vol(S(R))} \int _M \parallel \nabla ^M\sinh \,r \parallel ^2 \end{aligned}$$

where \(R>0\) is such that \(Vol(\varOmega ) = Vol(B(R))\); here, \(B(R)\) and \(S(R)\) are the geodesic ball and geodesic sphere, respectively, of radius \(R\). Further, the equality holds in above two inequalities if and only if \(M\) is a geodesic sphere of radius \(R\).

Proof

When \(k = 1\), the inequality (0.1) reduces to

$$\begin{aligned} \lambda _1(M)Vol(S(R))\sinh ^2R&\, \le \, (n-1)Vol(M) + \int _M \parallel \nabla ^M\sinh \,r \parallel ^2\,dm. \end{aligned}$$

Using the fact that \(\lambda _1(S(r)) = \frac{n-1}{\sinh ^2r}\) for all \(r>0\), we get the required result

$$\begin{aligned} \frac{\lambda _1(M)}{\lambda _1(S(R))} \le \frac{Vol(M)}{Vol(S(R))} + \frac{1}{(n-1)Vol(S(R))}\int _M \parallel \nabla ^M\sinh \,r \parallel ^2 \end{aligned}$$
(0.2)

for hypersurfaces in \(\mathbb {H}^n\).

When \(k > 1 \), we get

$$\begin{aligned} \lambda _1(M)&\le \left( \frac{kn-1}{\sinh ^2R} - \frac{k-1}{\cosh ^2R}\right) \frac{Vol(M)}{Vol(S(R))}\nonumber \\&\quad +\,\frac{1}{Vol(S(R))}\left( \frac{k-1}{\cosh ^2R}Vol(M) + \frac{1}{\sinh ^2R}\int _M \parallel \nabla ^M\sinh \,r \parallel ^2\right) \nonumber \\&= \lambda _1(S(R))\left( \frac{Vol(M)}{Vol(S(R))}\right) + \frac{k-1}{\cosh ^2R}\left( \frac{Vol(M)}{Vol(S(R))}\right) \nonumber \\&\quad +\, \ \frac{1}{\sinh ^2R\,Vol(S(R))} \int _M \parallel \nabla ^M\sinh \,r \parallel ^2. \end{aligned}$$
(0.3)

The equality in (0.2) and in (0.3) follows from the equality criterion in Lemmas 1 and 2 and \(\frac{\partial f_i}{\partial \eta }(q) = 0\) for all \( i = 1, \ldots , kn \) for all points \(q \in M\). This happens if and only if \(M\) is a geodesic sphere. \(\square \)