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
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 \(\overline{M}\),
where \(f=\sinh r\).
Substituting the value of \(\lambda _1(S(r)) \) and applying Lemmas 1 and 2 of [1], we get
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
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
and for \(k>1\), we have
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
Using the fact that \(\lambda _1(S(r)) = \frac{n-1}{\sinh ^2r}\) for all \(r>0\), we get the required result
for hypersurfaces in \(\mathbb {H}^n\).
When \(k > 1 \), we get
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. \(\square \)
References
Binoy, R., Santhanam, G.: Sharp upper bound for the first eigenvalue. Geometriae Dedicata 169(1), 397–410 (2014)
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The online version of the original article can be found under doi:10.1007/s10711-013-9863-0.
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Binoy, R., Santhanam, G. Erratum to: Sharp upper bound for the first eigenvalue. Geom Dedicata 174, 409–411 (2015). https://doi.org/10.1007/s10711-014-0023-y
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DOI: https://doi.org/10.1007/s10711-014-0023-y