Morse Index Bound for Minimal Two Spheres

Given a closed manifold of dimension at least three, with non-trivial homotopy group $$\pi _3(M)$$ π 3 ( M ) and a generic metric, we prove that there is a finite collection of harmonic spheres with Morse index bounded by one, with sum of their energies realizing a geometric invariant width.

Index estimates for closed minimal submanifolds of the sphere

In this paper we are interested in comparing the spectra of two elliptic operators acting on a closed minimal submanifold of the Euclidean unit sphere. Using an approach introduced by Savo in [A Savo. Index Bounds for Minimal Hypersurfaces of the Sphere. Indiana Univ. Math. J. 59 (2010), 823-837.], we are able to compare the eigenvalues of the stability operator acting on sections of the normal bundle and the Hodge Laplacian operator acting on $1$ -forms. As a byproduct of the technique and under a suitable hypothesis on the Ricci curvature of the submanifold we obtain that its first Betti's number is bounded from above by a multiple of the Morse index, which provide evidence for a well-known conjecture of Schoen and Marques & Neves in the setting of higher codimension.

Stationary Boltzmann equation: an approach via Morse theory

In this paper we study the unidimensional Stationary Boltzmann Equation by an approach via Morse theory. We define a functional J whose critical points coincide with the solutions of the Stationary Boltzmann Equation. By the calculation of Morse index of J’’0(0)h and the critical groups C2(J, 0) and C2(J, ∞) we prove that J has two different critical points u1 and u2 different from 0, that is, solutions of Boltzmann Equation.

Dual variational methods for a nonlinear Helmholtz equation with sign-changing nonlinearity

We prove new existence results for a nonlinear Helmholtz equation with sign-changing nonlinearity of the form $$\begin{aligned} - \Delta u - k^{2}u = Q(x)|u|^{p-2}u, \quad u \in W^{2,p}\left( {\mathbb {R}}^{N}\right) \end{aligned}$$ - Δ u - k 2 u = Q ( x ) | u | p - 2 u , u ∈ W 2 , p R N with $$k>0,$$ k > 0 , $$N \ge 3$$ N ≥ 3 , $$p \in \left[ \left. \frac{2(N+1)}{N-1},\frac{2N}{N-2}\right) \right. $$ p ∈ 2 ( N + 1 ) N - 1 , 2 N N - 2 and $$Q \in L^{\infty }({\mathbb {R}}^{N})$$ Q ∈ L ∞ ( R N ) . Due to the sign-changes of Q, our solutions have infinite Morse-Index in the corresponding dual variational formulation.

On the Morse index of branched Willmore spheres in 3-space

We develop a general method to compute the Morse index of branched Willmore spheres and show that the Morse index is equal to the index of certain matrix whose dimension is equal to the number of ends of the dual minimal surface (when the latter exists). As a corollary, we find that for all immersed Willmore spheres $$\vec {\Phi }:S^2\rightarrow \mathbb {R}^3$$ Φ → : S 2 → R 3 such that $$W(\vec {\Phi })=4\pi n$$ W ( Φ → ) = 4 π n , we have $$\mathrm {Ind}_{W}(\vec {\Phi })\le n-1$$ Ind W ( Φ → ) ≤ n - 1 .