\[ \begin{array} { r l } { ( 1 1 ) } & { \displaystyle \int _ { M } \left[ \sum _ { k = 1 } ^ { n } | I I ( e _ { k } , \omega ^ { \sharp } ) | ^ { 2 } + \sum _ { k = 1 } ^ { n } | I I ( e _ { k } , N ) | ^ { 2 } | \omega | ^ { 2 } \right] d M } \\ & { \displaystyle - \int _ { M } \left[ \sum _ { k = 1 } ^ { n } R m ^ { N } ( e _ { k } , \omega ^ { \sharp } , e _ { k } , \omega ^ { \sharp } ) + R i c ^ { N } ( N , N ) | \omega | ^ { 2 } \right] d M < \eta \int _ { M } | \omega | ^ { 2 } d M . } \end{array} \]

Then

\[ \# \{ e i g e n v a l u e s ~ o f ~ t h e ~ J a c o b i ~ o p e r a t o r ~ o f ~ M ^ { n } ~ t h a t ~ a r e < \eta \} \geq \frac { 2 } { d ( d - 1 ) } q . \]

Proof. (Compare [29] Theorem 16 and [30] Theorem 1.1). Let k be the number of eigenvalues of the Jacobi operator (5) of \( M ^ { n } \) that are below \( \eta \). Denote by \( \phi _ { 1 } , \ldots , \phi _ { k } \) the eigenfunctions associated to the k eigenvalues \( \lambda _ { 1 } \le \lambda _ { 2 } \le \lambda _ { 3 } \ldots \leq \lambda _ { k } \) of the Jacobi operator of \( M ^ { n } \) that are strictly smaller than \( \eta \).

Fix some global orthonormal basis \( \{ \theta _ { i j } \} _ { i < j } \) of \( \Lambda ^ { 2 } \mathbb { R } ^ { d } \) and let \( u _ { i j } \, = \)\( \langle N \wedge \omega ^ { \sharp } , \theta _ { i j } \rangle \) be the test functions defined in Proposition 2. The map that assigns to each \( \omega \in \mathcal { V } \) the vector

\[ \left[ \int _ { M } u _ { i j } \phi _ { p } d M \right] , \]

where \( i < j \) range from 1 to d and p ranges from 1 to k, is a linear map from the q dimensional vector space \( \nu \) to a vector space of dimension

\[ { \binom { d } { 2 } } k = { \frac { d ( d - 1 ) } { 2 } } k . \]

Assume, by contradiction, that \( \begin{array} { r } { q > \frac { d ( d - 1 ) } { 2 } k } \end{array} \). Then there would exist \( \omega \) in \( \mathcal { V } \setminus \{ 0 \} \) such that \( \begin{array} { r } { \int _ { M } u _ { i j } \phi _ { p } d M = 0 } \end{array} \) for all \( i < j \) and all p. Thus, as each \( u _ { i j } \) is \( L ^ { 2 } \)-orthogonal to all the first k eigenfunctions \( \phi _ { p } \), from the Courant-Hilbert variational characterization of eigenvalues it follows that

\[ \sum _ { i < j } ^ { d } Q ( u _ { i j } , u _ { i j } ) \ge \lambda _ { k + 1 } \sum _ { i < j } ^ { d } \int _ { M } u _ { i j } ^ { 2 } d M = \lambda _ { k + 1 } \int _ { M } | \omega | ^ { 2 } d M \ge \eta \int _ { M } | \omega | ^ { 2 } d M . \]

In view of Proposition 2, this is a contradiction with the assumption that inequality (11) holds for all \( \omega \) in \( \mathcal { V } \setminus \{ 0 \} \). The result follows.

q.e.d.

Remark 4.1. The same proof of the above proposition also gives more refined information when the two-sided immersion \( M ^ { n } \) arises as the two-sided cover of an one-sided immersed hypersurface in \( N ^ { n + 1 } \) (see sections

2.1 and 2.3). If one moreover assumes that all harmonic one-forms on the subspace \( \nu \) are such that all the corresponding functions \( u _ { i j } \) are odd with respect to the deck transformation of the cover, then one can repeat the argument considering the restriction of the Jacobi operator of \( M ^ { n } \) to the space of odd functions. It then follows that the number of eigenvalues of the one-sided hypersurface covered by \( M ^ { n } \) below the threshold \( \eta \) is bounded from below by 2 \( 2 q / d ( d - 1 ) \).

As a corollary, in the case \( \eta \) = 0 we obtain an estimate on the number of negative values of the Jacobi operator, that is, an index estimate:

Theorem A. Let \( ( \mathcal { N } ^ { n + 1 } , g ) \) be a Riemannian manifold that is iso-metrically embedded in some Euclidean space \( \mathbb { R } ^ { d } \). Let \( M ^ { n } \) be a closed embedded minimal hypersurface of \( ( \mathcal { N } ^ { n + 1 } , g ) \).

Assume that for every non-zero vector field X on \( M ^ { n } \),

\[ \begin{array} { l l } { \displaystyle \int _ { M } \left[ \sum _ { k = 1 } ^ { n } R m ^ { N } ( e _ { k } , X , e _ { k } , X ) + R i c ^ { N } ( N , N ) | X | ^ { 2 } \right] d M } \\ { \displaystyle } & { \displaystyle \qquad \qquad \> \int _ { M } \left[ \sum _ { k = 1 } ^ { n } | I I ( e _ { k } , X ) | ^ { 2 } + \sum _ { k = 1 } ^ { n } | I I ( e _ { k } , N ) | ^ { 2 } | X | ^ { 2 } \right] d M } \end{array} \]

Then

\[ ( 1 2 ) \qquad \qquad \qquad \qquad i n d e x ( M ) \geq \frac { 2 } { d ( d - 1 ) } b _ { 1 } ( M ) . \]

Proof. Assume \( M ^ { n } \) is two-sided. Under the assumption of the corol-lary, the hypothesis of Proposition 3 is automatically satisfied for \( \eta \)= 0 and \( \nu \) the set of all harmonic one-forms on \( M ^ { n } \), whose dimension is \( b _ { 1 } ( M ) \). Inequality (12) follows.

When \( M ^ { n } \) is one-sided, let \( \pi \, = \, \hat { M } ^ { n } \, \rightarrow \, M ^ { n } \, \subset \, \mathcal { N } ^ { n + 1 } \) be its two-sided cover and let V be the set of harmonic one-forms on \( \hat { M } ^ { n } \) that are invariant under the deck transformation \( \tau : \hat { M } ^ { n } \rightarrow \hat { M } ^ { n } \). This space has dimension at least \( b _ { 1 } ( M ) \), as it contains all the forms \( \pi ^ { * } \omega \), where \( \omega \) is harmonic on \( M ^ { n } \)(in fact, \( \pi ^ { * } : H ^ { 1 } ( M ; \mathbb { R } ) \to H ^ { 1 } ( \hat { M } ; \mathbb { R } ) \) is injective). The result follows as a consequence of Proposition 3 (see Remark 4.1) once one checks that, by construction of \( \hat { M } ^ { n } \), for all \( \hat { \omega } \) in V, each function \( \hat { u } _ { i j } = \langle \hat { N } \wedge \omega ^ { \sharp } , \theta _ { i j } \rangle \) satisfy

\[ \hat { u } _ { i j } ( \tau ( x ) ) = \langle \hat { N } ( \tau ( x ) ) \wedge \omega ^ { \sharp } ( \tau ( x ) ) , \theta _ { i j } \rangle = \langle - \hat { N } ( x ) \wedge \omega ^ { \sharp } ( x ) , \theta _ { i j } \rangle = - \hat { u } _ { i j } ( x ) \]

for all x in \( \hat { M } ^ { n } \), i.e, all functions \( u _ { i j } \) are odd with respect to the deck transformation \( \tau \).

q.e.d.