F-minimal Hypersurfaces Research Articles

Eigenvalue Estimates on Weighted Manifolds

We derive various eigenvalue estimates for the Hodge Laplacian acting on differential forms on weighted Riemannian manifolds. Our estimates unify and extend various results from the literature and provide a number of geometric applications. In particular, we derive an inequality which relates the eigenvalues of the Jacobi operator for f-minimal hypersurfaces and the spectrum of the Hodge Laplacian.

Index and first Betti number of f-minimal hypersurfaces: general ambients

We generalize a method by L. Ambrozio, A. Carlotto, and B. Sharp to study the Morse index of closed f-minimal hypersurfaces isometrically immersed in a general weighted manifold. The technique permits, in particular, to obtain a linear lower bound on the Morse index via the first Betti number for closed f-minimal hypersurfaces in products of some compact rank one symmetric spaces with an Euclidean factor, endowed with the rigid shrinking gradient Ricci soliton structure. These include, as particular cases, all cylindric shrinking gradient Ricci solitons.

The rigidity and stability of complete 𝑓-minimal hypersurfaces in ℝⁿ×𝕊¹(𝕒)

In this paper we consider f f -minimal hypersurfaces in the product space R n × S 1 ( a ) \mathbb {R}^n \times \mathbb {S}^1(a) where ( R n , e − f d μ ) (\mathbb {R}^n,e^{-f}d\mu ) is the standard Gaussian space with d μ d\mu being the standard volume element on R n \mathbb {R}^n . By introducing a globally defined smooth function α \alpha , we shall derive some interesting differential identities by which we are able to prove several rigidity theorems. We also study the stability properties of some standard examples. As a result, we prove that R n × { s 0 } ↪ R n × S 1 ( a ) \mathbb {R}^n \times \{s_0\} \hookrightarrow \mathbb {R}^n \times \mathbb {S}^1(a) is the only complete, proper, and stable f f -minimal hypersurface.

Harmonic 1-forms on immersed hypersurfaces in a Riemannian manifold with weighted Bi-Ricci curvature bounded from below

In this paper, we investigate Lp harmonic 1-forms on a complete noncompact orientable stable hypersurface that is immersed in a Riemannian manifold with weighted Bi-Ricci curvature bounded from below. We obtain several vanishing properties for this class of harmonic 1-forms. Our results can be considered as a generalization and improvement of previous work. Moreover, we consider immersed f-minimal hypersurfaces in a weighted Riemannian manifold and prove that if such a hypersurface is weighted stable then the space of L2 weighted harmonic 1-forms is trivial. As a consequence, any weighted stable minimal surface must have genus zero.

English

We give the definition of f-biminimal submanifolds and derive the equation for f-biminimal submanifolds. As an application, we give some examples of f-biminimal manifolds. Finally, we consider f-minimal hypersurfaces in the product space $$\mathbb{R}^{n} \times \mathbb{S}^{1}(a)$$ and derive two rigidity theorems.

Weighted volume growth and vanishing properties of [formula omitted]-minimal hypersurfaces in a weighted manifold

In this paper, we prove that a complete noncompact submanifold in a weighted manifold with nonpositive sectional curvature has at least linear weighted volume growth. Moreover we obtain several sufficient conditions for f-minimal hypersurfaces to have infinite weighted volume. By using an f-Laplacian comparison result, we obtain a lower bound of the first eigenvalue for the f-Laplace operator on submanifolds in a weighted manifold. We also obtain vanishing results for Lf2 harmonic 1-forms on complete noncompact f-minimal hypersurfaces in a weighted manifold. Finally we prove a topological structure theorem for complete noncompact Lf-stable f-minimal hypersurfaces via a Liouville-type theorem for f-harmonic functions with finite f-energy.

On the Stability and Parabolicity of Complete f-Minimal Hypersurfaces in Weighted Warped Products

Our purpose in this paper is to study the stability and parabolicity of f-minimal hypersurfaces immersed in a weighted warped product of the type $$I\times _{\rho }M^n_f$$ , where $$M^n_f$$ stands for a weighted manifold endowed with a weight function f. In this setting, we obtain sufficient conditions to guarantee that an f-minimal hypersurfaces immersed in $$I\times _{\rho }M^n_f$$ be $$L_f$$ -stable, where $$L_f$$ stands for the weighted Jacobi operator. Moreover, we establish a criterion of f-parabolicity and we apply it to infer when an f-minimal hypersurface immersed in $$I\times _{\rho }M^n_f$$ is totally geodesic. A study of the existence of f-minimal entire graphs in $$I\times _{\rho }M^n_f$$ is also made.

Stability and geometric properties of constant weighted mean curvature hypersurfaces in gradient Ricci solitons

In this paper, we study stability properties of hypersurfaces with constant weighted mean curvature (CWMC) in gradient Ricci solitons. The CWMC hypersurfaces generalize the f-minimal hypersurfaces and appear naturally in the isoperimetric problems in smooth metric measure spaces. We obtain a result about the relationship between the properness and extrinsic volume growth under the assumption of a limitation for the weighted mean curvature of the immersion. Moreover, we estimate Morse index for CWMC hypersurfaces in terms of the dimension of the space of parallel vector fields restricted to hypersurface.

Smooth compactness of 𝑓-minimal hypersurfaces with bounded 𝑓-index

Let ( M n + 1 , g , e − f d μ ) (M^{n+1},g,e^{-f}d\mu ) be a complete smooth metric measure space with 2 ≤ n ≤ 6 2\leq n\leq 6 and Bakry-Émery Ricci curvature bounded below by a positive constant. We prove a smooth compactness theorem for the space of complete embedded f f -minimal hypersurfaces in M M with uniform upper bounds on f f -index and weighted volume. As a corollary, we obtain a smooth compactness theorem for the space of embedded self-shrinkers in R n + 1 \mathbb {R}^{n+1} with 2 ≤ n ≤ 6 2\leq n\leq 6 . We also prove some estimates on the f f -index of f f -minimal hypersurfaces.

Lf-index bounds for closed f-minimal hypersurfaces

We discuss a two-side closed embedded f-minimal hypersurface Σ in a smooth metric measure space (Mn+1,g¯,e−fdμ) and obtain that Lf-index of Σ is bounded from below by a constant times the first Betti number with a suitable condition. We also apply the result to f-minimal hypersurfaces in shrinking gradient solitons.

Stability properties and gap theorem for complete f-minimal hypersurfaces

In this paper, we study complete oriented $f$-minimal hypersurfaces properly immersed in a cylinder shrinking soliton $(\mathbb{S}^n\times \mathbb{R}, \bar{g}, f)$. We prove that such hypersurface with $L_f$-index one must be either $\mathbb{S}^n\times\{0\}$ or $\mathbb{S}^{n-1}\times\mathbb{R}$, where $\mathbb{S}^{n-1}$ denotes the sphere in $\mathbb{S}^{n}$ of the same radius. Also we prove a pinching theorem for them.

Eigenvalue estimate and compactness for closedf-minimal surfaces

Let be a bounded domain with convex boundary in a complete noncompact Riemannian manifold with Bakry-´ Emery Ricci curvature bounded below by a positive constant. We prove a lower bound of the first eigenvalue of the weighted Laplacian for closed em- bedded f-minimal hypersurfaces contained in . Using this estimate, we prove a compactness theorem for the space of closed embedded f- minimal surfaces with the uniform upper bounds of genus and diameter in a complete 3-manifold with Bakry-´ Emery Ricci curvature bounded below by a positive constant and admitting an exhaustion by bounded domains with convex boundary.