The Frankel property for self-shrinkers from the viewpoint of elliptic PDEs

1.1 Weighted manifolds

By a weighted manifold (also called manifold with density, or smooth metric measure space) we mean a triad

where ( M , g ) is an m-dimensional Riemannian manifold with volume element dv,

and f : M → ℝ is a smooth weight function. The (obviously intrinsic) geometric analysis of M f is related to bounds on its Bakry–Émery Ricci curvature

combined with the analysis of its weighted Laplacian

where the weighted divergence is the operator

An important example of weighted manifold is represented by the Gaussian space

Since the weight function is f ⁢ ( x ) = 1 2 ⁢ | x | 2 , we have that the Bakry–Émery Ricci curvature is

hence ℝ f m + 1 is called a shrinking Ricci soliton. Moreover, the weighted Laplacian is the Ornstein–Uhlenbeck operator

1.2 Self shrinkers of the MCF

Given an isometrically immersed hypersurface in the weighted manifold M f m + 1

we introduce the corresponding weighted mean curvature vector field of the immersion as

where we are using the convention 𝐇 = tr Σ ⁡ 𝐀 , 𝐀 being the vector-valued second fundamental form. Here ( ⋅ ) ⊥ denotes the orthogonal projection on the normal bundle of Σ. We say that x : Σ m → M f m + 1 is f - minimal if 𝐇 f ≡ 0 .

A self-shrinker of the mean curvature flow (MCF) in the Euclidean space ℝ m + 1 is an f-minimal hypersurface of the Gaussian space ℝ f m + 1 . This is completely equivalent to require that the mean curvature vector field satisfies the equation

1.3 Intrinsic vs. extrinsic weighted structure

Clearly, the self-shrinker Σ m inherits the weighted structure of the ambient space. Thus, intrinsically, we can consider the manifold with density

where f ~ = f ∘ x . It is customary to drop the “tilde” in the weight function and to write Σ f m . An important relation between the (intrinsic) Bakry–Émery Ricci tensor of Σ f m and the extrinsic geometry of the f-minimal hypersurface Σ m comes from the Gauss equations. Indeed, it was observed in [22] that

As in the usual minimal surface theory, another important link between the intrinsic weighted geometry and the extrinsic properties of the self-shrinker comes from the f-Laplacian of the immersion. We have the following identity (see e.g. [4])

and its direct consequence

1.4 Properly immersed self-shrinkers

In this paper, we are mainly interested in properly immersed self-shrinkers. A remarkable result by Q. Ding and Y. L. Xin [5] states that a properly immersed self-shrinker x : Σ m → ℝ f m + 1 has extrinsic Euclidean volume growth

and, hence, finite weighted volume

This latter condition implies that the complete weighted manifold Σ f m is f - parabolic, i.e., for any u ∈ 𝒞 0 ⁢ ( Σ ) ∩ 𝒲 loc 1 , 2 ⁢ ( Σ ) ,

It is well known that parabolicity is a kind of compactness from several viewpoints, including global Stokes theorems and maximum principles, as it is already visible from the above definition.

1.5 The Frankel property

In many instances, properly immersed self-shrinkers behave like compact minimal hypersurfaces of the standard sphere. In this latter setting, it is well known that any two closed minimal immersed hypersurfaces must intersect. Actually, the ambient space can be generalized to a compact Riemannian manifold with strictly positive Ricci curvature. This is called the Frankel property after the celebrated paper by T. Frankel [6]. The original proof gives an estimate of the (positive) distance between non-intersecting compact hypersurfaces in terms of their mean curvatures and it is based on the second variation of length along a geodesic realizing the distance. New arguments and further extensions of the Frankel property to other geometric contexts are now available. Most notably, and relevantly for the development of the present paper, we mention [18] by P. Petersen and F. Wilhelm, where the maximum principle for the hypersurface-distance function is used, and [7] by A. Fraser and M. M.-C. Li, where the Frankel property is investigated in the setting of compact embedded free boundary minimal surfaces in a manifold with non-negative Ricci curvature. We shall comment on these works later on, in Section 3. It is natural to ask:

Starting from the work by G. Wei and W. Wylie, [26], where the case of compact hypersurfaces is considered (actually what is really needed is that the positive distance between the hypersurfaces is realized at finite points), few partial positive answers to this question appeared in the literature. They are mostly related to (generalized) half-space properties of the shrinkers; see Section 2.

During the summer school “Geometric Analysis on Riemannian and Singular Metric Measure Spaces”, held in Como in July 2016, http://arms.lakecomoschool.org, Professor Tom Ilmanen suggested (and kindly outlined the main steps of the parabolic proof) that the Frankel property can be proved in the general framework of the motion by level-sets in Euclidean space.

1.6 Purpose of the paper

In the present paper, by taking the purely elliptic viewpoint, we give the first general result in the literature about the validity of the (smooth) properly embedded Frankel property for self-shrinkers of the MCF. To this end, we shall collect results and techniques that, we feel, will be interesting also in other settings. The main contributions are the following:

1. With a new direct argument, based on the potential theory of weighted manifolds, we recover the main result of [1], namely we show that a properly immersed self-shrinker cannot be located neither inside nor outside a self-shrinker cylinder; see Theorem A in Section 2.

2. We apply a localized version of the Reilly formula to a suitable f-harmonic function with controlled gradient in order to show that two properly embedded self-shrinkers that are sufficiently separated at infinity must intersect at a finite point; see Theorem B and Theorem C in Section 3.

5.1 Uniqueness of the solution

Following [12], it is convenient to set the following:

Adapting to the framework of manifolds with density what is known from [12] (see also [11]), we have the following:

Since, in the setting of Lemma E, vol f ⁢ ( Ω ) ≤ vol f ⁢ ( M ) < + ∞ , the complete manifold Ω ¯ with boundary ∂ ⁡ Ω ≠ ∅ is f-parabolic in the sense of Dirichlet. Hence a bounded solution of (5.1), if any, must be unique.

5.2 Existence of the solution by exhaustion

As in the statement of Lemma E, let M f m + 1 be a complete smooth metric measure space and let Ω ¯ ⊂ M be the domain whose boundary is given by ∂ ⁡ Ω = Σ 1 ∪ Σ 2 . Let D k ↗ M be an exhaustion of M by relatively compact domains with smooth boundary ∂ ⁡ D k intersecting transversally Σ 1 and Σ 2 ; see e.g. [17]. Let Ω k ↗ Ω be the Lipschitz domains defined by Ω k = D k ∩ Ω . Note that

where Σ i , k ⊂ Σ i , i = 1 , 2 , and Γ k ⊂ ∂ ⁡ D k . The singular set of Ω k is denoted by 𝒮 k . Then { Ω k } is a “good” exhaustion of Ω with respect to mixed boundary value problems. Consider the solution u k to the problem

where ν k is the outward unit normal to Γ k . It follows from the Perron construction in [15], and the well-known local regularity theory, that u k ∈ 𝒞 0 ⁢ ( Ω ¯ k ) ∩ 𝒞 ∞ ⁢ ( Ω ¯ k ∖ 𝒮 k ) . Furthermore, by the strong maximum principle and the boundary point lemma,

For any fixed k 0 ∈ ℕ , let us consider the sequence of solutions of (5.3):

We claim that, given α > 0 , there exists a constant C k 0 > 0 such that

To this end:

1. We apply [8, Corollary 6.7] with the choices Ω := Ω k 0 + 1 , T := Σ 1 , k 0 + 1 , L := Δ f , φ := 0 , f := 0 and we obtain that there exist a ray δ 1 , k 0 > 0 and a constant C 1 , k 0 > 0 such that, having defined the δ k 0 -neighborhood of Σ 1 , k 0 as

   𝒯 δ 1 , k 0 ⁢ ( Σ 1 , k 0 ) = ⋃ p ∈ Σ 1 , k 0 B δ 1 , k 0 M ⁢ ( p ) ∩ Ω k 0 + 1 ,

   it holds

   ∥ u k ∥ 𝒞 2 , α ⁢ ( 𝒯 δ 1 , k 0 ⁢ ( Σ 1 , k 0 ) ) ≤ C 1 , k 0

   for every u k ∈ 𝒰 k 0 + 2 .

2. Similarly, we apply [8, Corollary 6.7] with the choices Ω := Ω k 0 + 1 , T := Σ 2 , k 0 + 1 , L := Δ f , φ := 1 , f := 0 and we obtain that there exist a ray δ 2 , k 0 > 0 and a constant C 2 , k 0 > 0 such that

   ∥ u k ∥ 𝒞 2 , α ⁢ ( 𝒯 δ 2 , k 0 ⁢ ( Σ 2 , k 0 ) ) ≤ C 2 , k 0

   for every u k ∈ 𝒰 k 0 + 2 .

3. Set

   δ k 0 = min ⁡ ( δ 1 , k 0 , δ 2 , k 0 ) > 0

   and define the compact set

   𝒦 k 0 = Ω ¯ k 0 ∖ ( 𝒯 δ k 0 2 ⁢ ( Σ 1 , k 0 ) ∪ 𝒯 δ k 0 2 ⁢ ( Σ 2 , k 0 ) ) ⊂ Ω k 0 + 1 .

   By [8, Theorem 6.2] with Ω := Ω k 0 + 1 and L := Δ f , there exists a constant C 3 , k 0 > 0 such that

   ∥ u k ∥ 𝒞 2 , α ⁢ ( 𝒦 k 0 ) ≤ C 3 , k 0

   for every u k ∈ 𝒰 k 0 + 2 .

4. Since

   Ω k 0 ⊂ 𝒯 δ k 0 ⁢ ( Σ 1 , k 0 ) ∪ 𝒯 δ k 0 ⁢ ( Σ 2 , k 0 ) ∪ 𝒦 k 0   and   𝒰 k 0 + 2 ⊂ 𝒞 ∞ ⁢ ( Ω ¯ k 0 + 1 ) ,

   the claimed estimate (5.4) follows by taking C k 0 = max ⁡ ( C 1 , k 0 , C 2 , k 0 , C 3 , k 0 ) .

Now, for 0 < α 1 < α 2 , the embedding 𝒞 2 , α 2 ⁢ ( Ω ¯ k 0 ) ↪ 𝒞 2 , α 1 ⁢ ( Ω ¯ k 0 ) is compact. Therefore, possibly passing to a subsequence, we obtain that 𝒰 k 0 + 2 converges in 𝒞 2 ⁢ ( Ω ¯ k 0 ) to a solution u k 0 ∈ 𝒞 2 ⁢ ( Ω ¯ k 0 ) (actually u k 0 ∈ 𝒞 ∞ ⁢ ( Ω ¯ k 0 ) by higher elliptic regularity) of the problem

Moreover,

To conclude the construction, we let k 0 increase to + ∞ and we use a classical diagonal argument. This yields the desired solution u ∈ 𝒞 ∞ ⁢ ( Ω ¯ ) of

satisfying

5.3 The finite f-energy condition

From now on, unless otherwise specified, we assume that M f is the Gaussian space ℝ f m + 1 . We want to obtain the finiteness of the Dirichlet f-energy of the solution u of (5.1): ∥ ∇ ⁡ u ∥ ℒ 2 ⁢ ( Ω , d ⁢ v f ) < + ∞ . A standard Caccioppoli inequality up to the boundary reduces the problem to an estimate of | ∇ ⁡ u | along Σ 2 (the boundary hypersurface where the datum 1 is imposed). This latter, in turn, can be carried out by maximum principle considerations.