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.

Boundary separated and clustered layer positive solutions for an elliptic Neumann problem with large exponent

Given a smooth bounded domain [Formula: see text] in [Formula: see text] with [Formula: see text], we study the existence and the profile of positive solutions for the following elliptic Neumann problem: [Formula: see text] where [Formula: see text] is a large exponent and [Formula: see text] denotes the outer unit normal vector to the boundary [Formula: see text]. For suitable domains [Formula: see text], by a constructive way we prove that, for any non-negative integers [Formula: see text], [Formula: see text] with [Formula: see text], if [Formula: see text] is large enough, such a problem has a family of positive solutions with [Formula: see text] boundary layers and [Formula: see text] interior layers which concentrate along [Formula: see text] distinct [Formula: see text]-dimensional minimal submanifolds of [Formula: see text], or collapse to the same [Formula: see text]-dimensional minimal submanifold of [Formula: see text] as [Formula: see text].

On the geometry of submanifolds in certain warped products

In this paper, we deal with [Formula: see text]-dimensional submanifolds immersed in a slab of a warped product of the type [Formula: see text]. Under suitable constraints on the warping function [Formula: see text] and assuming that such a submanifold [Formula: see text] is either complete or stochastically complete, we apply some maximum principles in order to show that [Formula: see text] must be contained in a slice of [Formula: see text]. In particular, from our results we guarantee the nonexistence of [Formula: see text]-dimensional closed minimal submanifolds immersed in [Formula: see text]. Furthermore, we construct a nontrivial duo-graph in [Formula: see text] which illustrates the importance of our rigidity results.

The minimality of determinantal varieties

The determinantal variety {\Sigma_{pq}} is defined to be the set of all {p\times q} real matrices with {p\geq q} whose ranks are strictly smaller than q. It is proved that {\Sigma_{pq}} is a minimal cone in {\mathbb{R}^{pq}} and all its strata are regular minimal submanifolds.

Minimal submanifolds from the abelian Higgs model

Given a Hermitian line bundle $$L\rightarrow M$$ L → M over a closed, oriented Riemannian manifold M, we study the asymptotic behavior, as $$\epsilon \rightarrow 0$$ ϵ → 0 , of couples $$(u_\epsilon ,\nabla _\epsilon )$$ ( u ϵ , ∇ ϵ ) critical for the rescalings $$\begin{aligned} E_\epsilon (u,\nabla )=\int _M\Big (|\nabla u|^2+\epsilon ^2|F_\nabla |^2+\frac{1}{4\epsilon ^2}(1-|u|^2)^2\Big ) \end{aligned}$$ E ϵ ( u , ∇ ) = ∫ M ( | ∇ u | 2 + ϵ 2 | F ∇ | 2 + 1 4 ϵ 2 ( 1 - | u | 2 ) 2 ) of the self-dual Yang–Mills–Higgs energy, where u is a section of L and $$\nabla $$ ∇ is a Hermitian connection on L with curvature $$F_{\nabla }$$ F ∇ . Under the natural assumption $$\limsup _{\epsilon \rightarrow 0}E_\epsilon (u_\epsilon ,\nabla _\epsilon )<\infty $$ lim sup ϵ → 0 E ϵ ( u ϵ , ∇ ϵ ) < ∞ , we show that the energy measures converge subsequentially to (the weight measure $$\mu $$ μ of) a stationary integral $$(n-2)$$ ( n - 2 ) -varifold. Also, we show that the $$(n-2)$$ ( n - 2 ) -currents dual to the curvature forms converge subsequentially to $$2\pi \Gamma $$ 2 π Γ , for an integral $$(n-2)$$ ( n - 2 ) -cycle $$\Gamma $$ Γ with $$|\Gamma |\le \mu $$ | Γ | ≤ μ . Finally, we provide a variational construction of nontrivial critical points $$(u_\epsilon ,\nabla _\epsilon )$$ ( u ϵ , ∇ ϵ ) on arbitrary line bundles, satisfying a uniform energy bound. As a byproduct, we obtain a PDE proof, in codimension two, of Almgren’s existence result for (nontrivial) stationary integral $$(n-2)$$ ( n - 2 ) -varifolds in an arbitrary closed Riemannian manifold.