Upper bounds for the first Dirichlet eigenvalue of a tube around an algebraic complex curve of \(\mathbb C\mathbb{P}^{n}(\lambda)\) (Q636028)

Let \(\mathbb C\mathbb{P}^{n}(\lambda)\) be complex projective space with holomorphic sectional curvature \(\lambda\). Let \(P\) be a complete complex curve in \(\mathbb C\mathbb{P}^{n}(\lambda)\); \(P\) is algebraic by Chow's theorem and can be regarded as the set of zeros of \(n-1\) homogeneous polynomials of degrees \(a_2,\dots,a_n\). Let \(P_\rho\) be the tube of radius \(p\geq0\) about \(P\). Let \(\mathbb C\mathbb{P}^1(\lambda)\) be embedded as a complex totally geodesic submanifold of \(\mathbb C\mathbb{P}^n(\lambda)\). The authors show \smallbreak Theorem 1.1: The first eigenvalue \(\mu_1(P_\rho)\) of the Dirichlet Laplacian satisfies the inequality: \[ \mu_1(P_\rho)\leq\mu_1(\mathbb C\mathbb{P}^1(\lambda)_\rho)- {{2\lambda(\sum_{s=2}^na_s-(n-1))}\over {1-\lambda((n-1)^{-1}\sum_{s=2}^na_s)C(\rho)}} \] where \(C(\rho)\) satisfies \(\lambda((n-1)^{-1}\sum_{s=2}^na_s)C(\rho)<1\). Furthermore, the equality is attained iff \(P=\mathbb C\mathbb{P}^1(\lambda)\). \medbreak This bound relates the first Dirichlet eigenvalue of \(P_\rho\) with the degrees of the polynomials defining \(P\). This in turn is related with the first Chern class of the normal bundle of \(P\). One has a gap phenomenon for \(\mu_1(P_\rho)\) between the case \(P=\mathbb C\mathbb{P}^1(\lambda)\) (which corresponds to \(a_2=\dots=a_n=1\)) and other complex curves \(P\); the gap is measured by the degrees \(\{a_s\}\).