Stability and compactness for complete $f$-minimal surfaces

Let $(M,\bar{g}, e^{-f}d\mu)$ be a complete metric measure space with Bakry-\'Emery Ricci curvature bounded below by a positive constant. We prove that, in $M$, there is no complete two-sided $L_f$-stable immersed $f$-minimal hypersurface with finite weighted volume. Further, if $M$ is a 3-manifold, we prove a smooth compactness theorem for the space of complete embedded $f$-minimal surfaces in $M$ with the uniform upper bounds of genus and weighted volume, which generalizes the compactness theorem for complete self-shrinkers in $\mathbb{R}^3$ by Colding-Minicozzi.


Introduction
Recall that a self-shrinker (for mean curvature flow in R n+1 ) is a hypersurface Σ immersed in the Euclidean space (R n+1 , g can ) satisfying that where x is the position vector in R n+1 , ν is the unit normal at x, and H is the mean curvature curvature of Σ at x. Self-shrinkers play an important role in the study of singularity of mean curvature flow and have been studied by many people in recent years. We refer to [4], [5] and the references therein. In particular, Colding-Minicozzi [4] proved the following compactness theorem for self-shrinkers in R 3 .

Theorem 1. [4]
Given an integer g ≥ 0 and a constant V > 0, the space S(g, V ) of smooth complete embedded self-shrinkers Σ ⊂ R 3 with • genus at most g, • ∂Σ = ∅, • Area(B R (x 0 ) ∩ Σ) ≤ V R 2 for all x 0 ∈ R 3 and R > 0 2000 Mathematics Subject Classification. Primary: 58J50; Secondary: 58E30. The first and third authors are partially supported by CNPq and Faperj of Brazil. The second author is supported by CNPq of Brazil. is compact.
Namely, any sequence of these has a subsequence that converges in the topology of C m convergence on compact subsets for any m ≥ 2.
In this paper, we extend Theorem 1 to the space of complete embed- • f ≡ C, an f -minimal hypersurface is just a minimal hypersurface; • self-shrinker Σ in R n+1 . f = |x| 2 4 ; • Let (M, g, f ) be a shrinking gradient Ricci solitons, i.e. after a normalization, (M, g, f ) satisfies the equation Ric + ∇ 2 f = 1 2 g or equivalently the Bakry-Émery Ricci curvature Ric f := Ric + ∇ 2 f = 1 2 . We may consider f -minimal hypersurfaces in (M, g, f ). In particular, the previous example: a self-shrinker Σ in R n+1 is f -minimal in Gauss shrinking soliton (R n+1 , g can , |x| 2 4 ); • M = H n+1 (−1), the hyperbolic space. Let r denote the distance function from a fixed point p ∈ M and f (x) = nar 2 (x), where a > 0 is a constant. Now Ric f ≥ n(2a − 1). The geodesic sphere of radius r centered at p is an f -minimal hypersurface if the radius r satisfies 2ar = coth r.
An f -minimal hypersurface Σ can be viewed in two ways. One is that Σ is f -minimal if and only if Σ is a critical point of the weighted volume functional e −f dσ, where dσ is the volume element of Σ. The other one is that Σ is f -minimal if and only if Σ is minimal in the new conformal metricg = e − 2f n g (see Section 2 and Appendix). f -minimal hypersurfaces have been studied before, even more general stationary hypersurfaces for parametric elliptic functionals, see for instance the work of White [14] and Colding-Minicozzi [7].
We prove the following compactness result: Theorem 2. Let (M 3 , g, e −f dµ) be a complete smooth metric measure space and Ric f ≥ k, where k is a positive constant. Given an integer g ≥ 0 and a constant V > 0, the space S g,V of smooth complete embedded f -minimal surfaces Σ ⊂ M with • genus at most g, is compact in the C m topology, for any m ≥ 2. Namely, any sequence of S g,V has a subsequence that converges in the C m topology on compact subsets to a surface in S D,g , for any m ≥ 2.
Since the existence of the uniform scale-invariant area bound is equivalent to the existence of the uniform bound of the weighted area for self-shrinkers (see Remark 1 in Section 5), Theorem 2 implies Theorem 1. Also, in [2], we will apply Theorem 2 to obtain a compactness theorem for the space of closed embedded f -minimal surfaces with the upper bounds of genus and diameter.
To prove Theorem 2, we need to prove a non-existence result on L f -stable f -minimal hypersurfaces, which is of independent interest. Here we explain briefly the meaning of L f stability. For an f -minimal hypersurface Σ, L f operator is where ∆ f = ∆ − ∇f, ∇· is the weighted Laplacian on Σ.
Especially, for self-shrinkers, it is so-called L operator: L f -stability of Σ means that its weighted volume Σ e −f dσ is locally minimal, that is, the second variation of its weighted volume is nonnegative for all compactly supported normal variation. We leave more details about the definition of L f -stability and some of its properties in Section 2 and Appendix.
Since the first and third authors [3] of the present paper proved that for self-shrinkers, properness, the polynomial volume growth, and finite weighted volume are equivalent. Hence Theorem 3 implies Theorem 4.
In this paper, we discuss the relation among the properness, polynomial volume growth and finite weighted volume of f -minimal submanifolds (Propositions 3, 4 and 5). We obtain their equivalence when the ambient space (M, g, f ) is a shrinking gradient Ricci solitons, i.e. Ric + ∇ 2 f = 1 2 g with the condition |∇f | 2 ≤ f (Corollary 1).
The rest of this paper is organized as follows: In Section 2 some definitions, notations and facts are given as a preliminary; In Section 3 we prove Propositions 3, 4 and 5; In Section 4 we prove Theorem 3; In Section 5 we prove Theorem 2; In Appendix we discuss some properties of L f -stability for f -minimal submanifolds.

Preliminaries
In general, a smooth metric measure space, denoted by (M m , g, e −f dµ), is an m-dimensional Riemannian manifold (M m , g) together with a weighted volume form e −f dµ on M , where f is a smooth function on M and dµ the volume element induced by the metric g. In this paper, unless otherwise specified, we denote by a bar all quantities on (M, g), for instance by ∇ and Ric, the Levi-Civita connection and the Ricci curvature tensor of (M, g) respectively. For (M, g, e −f dµ), an important and natural tensor is the ∞-Bakry-Émery Ricci curvature tensor Ric f (for simplicity, Bakry-Émery Ricci curvature), which is defined by where instance, see the work of Wei-Wylie [13], Munteanu-Wang [11,12] and the references therein. In this paper, we will use the following proposition by Morgan [10] (see also its proof in [13] ). Then i : (Σ n ; i * g) → (M m , g) is an isometric immersion with the induced metric i * g. For simplicity, we still denote i * g by g whenever there is no confusion. We will denote for instance by ∇, Ric, ∆ and dσ, the Levi-Civita connection, the Ricci curvature tensor, the Laplacian, and the volume element of (Σ, g) respectively.
The function f induces a weighted measure e −f dσ on Σ. Thus we have an induced smooth metric measure space (Σ n , g, e −f dσ).
The associated weighted Laplacian ∆ f on (Σ, g) is defined by The second order operator ∆ f is a self-adjoint operator on the space of square integrable functions on Σ with respect to the measure e −f dσ (however the Laplacian operator in general has no this properties).
The second fundamental form A of (Σ, g) is defined by where ⊥ denotes the projection to the normal bundle of Σ. The mean Definition 1. The weighted mean curvature vector of Σ with respect to the metric g is defined by (1) The immersed submanifold (Σ, g) is called f -minimal if its weighted mean curvature vector H f vanishes identically, or equivalently if its mean curvature vector satisfies Definition 2. The weighted volume of (Σ, g) is defined by It is well known that Σ is f -minimal if and only if Σ is a critical point of the weighted volume functional. Namely, it holds that Proposition 2. If T is a compactly supported variational field on Σ, then the first variation formula of the weighted volume of (Σ, g) is given by On the other hand, an f -minimal submanifold can be viewed as a minimal submanifold under a conformal metric. Precisely, define the new metric g = e − 2 n f g on M , which is conformal to g. Then the immersion i : Σ → M induces a metric i * g on Σ from (M,g). In the following, i * g is still denoted byg for simplicity. The volume of (Σ,g) is Hence Proposition 2 and (5) imply that where dσ = e −f dσ andH denote the volume element and the mean curvature vector of Σ with respect to the conformal metricg respectively.
and ν a unit normal at p. The second fundamental form A and the mean curvature H of (Σ, g) are as follows: Hence the mean curvature vector H of (Σ, g) satisfies H = −Hν. Define the For a hypersurface (Σ, g), the L f operator is defined by where |A| 2 denotes the square of the norm of the second fundamental form The L f -stability of Σ is defined as follows: It is known that an f -minimal hypersurface (Σ, g) is L f -stable if and only if (Σ,g) is stable as a minimal surface with respect to the conformal metric g = e −f g. See more details in Appendix of this paper.
In this paper, for closed hypersurfaces, we choose ν to be the outer unit normal.
3. Properness, polynomial volume growth and finit weighted volume of f -minimal hypersurfaces In [3], the first and third authors of the present paper proved that the finite weighted volume of a self-shrinker Σ n immersed in R m implies it is properly immersed. In [9], Ding-Xin proved that a properly immersed self-shrinker must have the Euclidean volume growth. Combining these two results, it was proved [3] that for immersed self-shrinkers, properness, polynomial volume growth and finite weighted volume are equivalent.
In this section we study the relation among the properness, polynomial volume growth and finite weighted volume of f -minimal submaifolds, some of them will be used later in this paper.
If Σ is an n-dimensional submanifold in a complete manifold M m , n < m, Σ is said to have polynomial volume growth if, for a p ∈ M fixed, there exist constants C and d so that for all r ≥ 1, where B M r (p) is the extrinsic ball of radius r centered at p, Vol(B M r (p) denotes the volume of B M r (p) ∩ Σ. When d = n in (9), Σ is said to be of Euclidean volume growth.
Before proving the following Proposition 3, we recall an estimate implied by the Hessian comparison theorem (cf, for instance, [6] Lemma 7.1). Lemma 1. Let (M, g) be a complete Riemannian manifold with bounded geometry, that is, M has sectional curvature bounded by k (|K M | ≤ k), and injectivity radius bounded below by i 0 > 0. Then the distance function r(x) Using this estimate we will prove that Proof. We argue by contradiction. Since the argument is local, we may assume that (M, g) has bounded geometry. Suppose that Σ is not properly denotes an extrinsic ball of radius R centered at o. Then for any a > 0, there is a sequence from p j satisfies, Choosing a ≤ min{ n 2c , 2R}, we have for 0 where ν denotes the outward unit normal vector of ∂B Σ µ (p j ) and A(µ) denotes the area of ∂B Σ µ (p j ). Using co-area formula in (10), we have This implies Thus we conclude This contradicts with the assumption of the finite weighted volume of Σ. Proof. Since (M, g, f ) is a gradient shrinking Ricci soliton, it is well-known that, by a scaling of the metric g and a translating of f , still denoted by g and f , we may normalize the metric so that k = 1 2 and the following identities hold: From these equations, we have that It was proved by Cao and the third author [1] that there is a positive constant c so that for any x ∈ M with r(x) = dist M (p, x) ≥ r 0 , where p is an fixed point in M and c, r 0 are positive constants that depend only of m and f (p).
By (14), we know that f is a proper function on M . Since Σ is properly immersed in M and f is proper in M , f | Σ is also a proper smooth function on Σ. Note that with the scaling metric and translating f , Σ is still f -minimal. Hence Also we have  (14), we have that Σ has the Euclidean volume growth.
We prove the following 2kf . If Σ n is a complete submanifold (not necessarily f -minimal) with polynomial area growth, then Σ has finite weighted volume.
Proof. By a scaling of the metric, we may assume that k = 1 2 . The proof follows from the estimate of f . Munteanu-Wang [11] extended the estimate (14) to (M m , g, e −f dµ) with Ric f ≥ 1 2 and |∇f | 2 ≤ f . Combining the assumption that Σ has polynomial volume growth with the estimative (14), By Propositions 3, 4 and 5, we have the following

Non-existence of L f stabe f -minimal hypersurfaces
In this section, we prove Theorem 3, which is a key to prove the compactness theorem in Section 5. Proof. We argue by contradiction. Suppose that Σ is an L f -stable complete f -minimal hypersurface immersed in (M, g) without boundary and with finite weighted volume. Recall that a two-sided hypersurface Σ is L f -stable if the following inequality holds that, for any compactly supported smooth Observe that any closed hypersurface cannot be L f -stable. This is because that: the assumption Ric f ≥ k > 0 implies that (15) cannot hold for ϕ ≡ c on Σ. Hence, Σ must be noncompact.
Fix a point p ∈ Σ and let r(x) = dist Σ (p, x) denote the (intrinsic) distance function in Σ. Define a sequence of functions ϕ j (x) = η( r(x) j ), j ≥ 1. Then |∇ϕ j | 2 ≤ 1 for j ≥ 2. Substituting ϕ j for ϕ in (15): where B Σ j (p) is the intrinsic geodesic ball in M of radius j centered at p. Since Σ has finite weighted volume, we have, when j → ∞ Choosing j large enough, we have that ϕ j satisfies This contradicts that Σ is L f -stable.

Compactness of complete f -minimal surfaces
Before proving Theorem 2, we give some facts. if at least one of them is compact, there is a minimal geodesic joining Σ 1 and Σ 2 and realizing their distance. Hence the proof of Theorem 7.3 [13] can be applied to obtain the following Proposition 6. Let (M, g, e −f dµ) be an (n + 1)-dimensional smooth metric measure space with Ric f ≥ k, where k is a positive constant. If Σ 1 and Σ 2 are two complete immersed hypersurfaces, at least one of them is compact, where C is independent on Σ. Therefore there is a closed ball B M of M with radius big enough so that any Σ must intersect it.
We need the following fact: Proposition 7. Let M be a simply connected Riemannian manifold. If an f -minimal hypersurface Σ is complete, not necessarily connected, properly embedded, and has no boundary, then every component of Σ separates M into two components and thus is two-sided. Therefore Σ has a globally defined unit normal.
Proof. Suppose Σ j is a component of Σ. By contrary, if M \Σ j has one component. Since Σ is a properly embedded f -minimal hypersurface, for any p ∈ Σ j , there is a neighborhood W of p in M so that W ∩ Σ j = W ∩ Σ only has one piece (i.e. it is a graph above a connected domain in the tangent plane of p). Thus we have a simply closed curve γ passing p, transversal to Σ j at p, and Σ j ∩ γ = p. Since M is simply connected, we have a disk D with the boundary γ. Again since Σ is proper, the intersection of Σ j with ∂D = γ cannot be one point, which is a contradiction.
Combining Proposition 3 in Section 3 with Proposition 7, we obtain Proposition 8. Let (M, g, e −f dµ) be a simply connected complete smooth measure space. If a complete f -minimal hypersurface has finite weighted volume, then every component of Σ separates M into two components and thus is two-sided. Therefore Σ has a globally defined unit normal.
We will take the same approach as in Colding-Minicozzi's paper [4] to prove Theorem 2, a smooth compactness theorem for complete f -minimal surfaces. First we recall a well known local singular compactness theorem for embedded minimal surfaces in a Riemannian 3-manifold.
Also, it is clear that the genus ofB 2R (p) ∩ Σ j remains at most g. Then by Proposition 9, there exists a finite collection of points x k , a smooth embedded minimal surface Σ ⊂B R (p), with ∂Σ ⊂ ∂B R and a subsequence of {Σ j } that converges inB R (p) (with finite multiplicity) to Σ away from the set {x k }.
Let {B R i (p i )} be a countable cover of (M,g). On eachB 2R i (p i ), applying the previous local convergence, and then passing to a diagonal subsequence, we obtain that there is a subsequence of Σ i , still denoted by Σ i , a smooth embedded minimal surface Σ (with respect to the metricg) without boundary, and a locally finite collection of points S ⊂ Σ so that Σ i converges smoothly (possibly with multiplicity) to Σ off of S.
Since Σ has no boundary, it is complete in the original metric g. Thus we obtain that the smooth convergence of the subsequence to the smooth embedded complete f -minimal surface Σ off of S.
The convergence of Σ i to Σ and (17) imply Σ e −f dσ ≤ V . By Proposition 3, Σ is properly embedded.
We need to show that the convergence is smooth across the points in S.
To prove it, we need the following Proposition 11. Assume that the ambient manifold M in Proposition 10 is simply connected. If the convergence of the sequence {Σ i } has the multiplicity greater than one, then Σ is L f -stable.
Proof. By Proposition 8, we know that Σ i and Σ are orientable. We may have two ways to prove the proposition. The first is to use the known fact on minimal surfaces. It is known that (cf [6] Appendix A) if the multiplicity of the convergence of a sequence of embedded orientable minimal surfaces in a simply connected 3-manifold is not one, then the limit minimal surface is stable. Under the conformal metricg, a sequence {Σ i } of minimal surfaces converges to a smooth embedded orientable minimal surface Σ and thus Σ is stable. Besides, the conclusion that Σ is stable with respect to the conformal metricg is equivalent to that Σ is L f -stable under the original metric g (see Appendix).
The second way is to prove directly. We may prove that L f is the linearization of f -minimal equation by a similar proof to the one in [5,8] Appendix A. By arguing as in Proposition 3.2 in [4,8],, we can find a smooth positive function u on Σ satisfying This implies that Σ is L f -stable.
Proof of theorem 2. By the assumption on Ric f and Proposition 1, M has finite fundamental group. After passing to the universal covering, we may assume that M is simply connected. Given a sequence of smooth complete embedded f -minimal surfaces {Σ i } with genus g, ∂Σ i = ∅, and the weighted area at most V , by Proposition 10, there is a subsequence, still denoted by {Σ i } so that it converges in the topology of smooth convergence on compact subsets to a smooth embedded complete f -minimal surface Σ away from a locally finite set S ⊂ Σ (possibly with multiplicity). Moreover, the limit surface Σ ⊂ M is complete, properly embedded, Σ e −f dσ ≤ V , has no boundary and has a well-defined unit normal ν. We also have the equivalent convergence under the conformal metric g.
If S is not empty, Allard's regularity theorem implies that the convergence has multiplicity greater than one. Then by Proposition 11, we conclude that Σ is L f -stable. But Proposition 5 says that there is no such Σ. This contradiction implies that S must be empty. We complete the proof of the theorem. When (Σ,g) is minimal, it is well known that the second variation of the volume of (Σ,g) is given by Proposition 12. (cf [6]) Let (Σ,g) be minimal submanifold in (M,g). If T is a normal compactly supported variational vector field on Σ (that is, T = T ⊥ ), then the second variational formula of the volumeṼ of (Σ,g) is given by where the stability operator (or Jacobi operator) J is defined on a normal vector field T to Σ by Here is the Laplacian determined by the normal connection ∇ ⊥ of (Σ,g), Rm is the curvature tensor on (M,g), Ã (ẽ i ,ẽ j ), T Ã (ẽ i ,ẽ j ), and {ẽ i }, i = 1, · · · , n is a local orthonormal base of (Σ,g).
Recall that the weighted volume of (Σ, g) is defined by By a direct computation similar to that of (20), we may prove the second variational formula of the weighted volume of f -minimal submanifold (Σ, g).

Definition 5.
For any normal vector field T on (Σ, g), the second order operator ∆ ⊥ f is defined by The operator L f on (Σ, g) is defined by, In the above, ∇ ⊥ denotes the normal connection of (Σ,g); {e i }, i = 1, . . . , n is a local orthonormal base of (Σ, g); B(T ) = n i,j=1 where A denotes the second fundamental form of (Σ, g); [Rm(e i , T )e i ] ⊥ , where Rm denotes the Riemannian curvature tensor of (M, g); and is a local orthonormal normal vector field on (Σ, g).
Since ∂J ∂t = n i,j=1 g ij ∇ e i T, e j J, ∂J f ∂t = n i,j=1 g ij ∇ e i T, e j − ∇f, T J f . Note that T is a normal vector field. A direct computation gives, at (p, 0) Using div Observe that the right-hand side of (25) is independent of the choice of coordinates. Hence (25) holds on Σ. By integrating (25) and using the fact that Σ is f -minimal (i.e., H f = 0), we obtain Substituting Thus we have the second variational formula of the weighted volume of Σ Definition 6. An f -minimal submanifold (Σ, g) is called L f -stable if the second variational of the weighted volume of Σ given by (24) is nonnegative for any normal compactly supported variational vector field T on Σ.
Observe that for an f -minimal submanifold Σ and its normal compactly .
By (20)  We define the L f -index, denoted by L f -ind, of (Σ, g) by the maximum of the dimensions of negative definite subspaces of B f . Hence (Σ, g) is L f -stable if and only if its L f -ind= 0.
On the other hand, for minimal (Σ,g), it is well known that the stability operator J also defines a symmetric bilinear formB(T, T ), There are also the concepts of index and stability of (Σ,g). In particular, (Σ,g) is stable if and only if the index ind(Σ,g) = 0. Since B f (T, T ) =

B(T, T ), it holds that
Proposition 14. L f -ind of (Σ, g) is equal to the index of (Σ,g). In particular, (Σ, g) is L f -stable if and only if (Σ,g) is stable in (M,g).
Now if Σ is a two-sided hypersurface, that is, there is a globally-defined unit normal ν on (Σ, g). Take T = ϕν. Then the second variation (24) implies that Proposition 15. Let Σ be a two-sided f -minimal hypersurface in (M n+1 , g).
If ϕ is a compactly supported smooth function on Σ, then the second variation of the weighted volume of (Σ, g) is given by where ν denotes the unit normal of (Σ, g) and the operator L f is defined by L f = ∆ f + |A| 2 g + Ric f (ν, ν).
A bilinear form on space C ∞ o (Σ) of compactly supported smooth functions on Σ is defined by The L f -index, denoted by L f -ind, of (Σ, g) is defined to be the maximum of the dimensions of negative definite subspaces of B f . Hence (Σ, g) is Also, for minimal hypersurface i : (Σ,g) → (M n+1 ,g), it is well known that if ψ is a compactly supported smooth function on Σ, then the second variational of the volumeṼ of (Σ, i * g ) is given by whereÃ denotes the second fundamental form of (Σ,g),ν denotes the unit normal of (Σ,g), and J = △g + |Ã| 2 g + Ric(ν,ν) is stability operator (or the Jacobi operator) of (Σ,g).
It holds that, from (28),  Corollary 5. L f -ind of (Σ, g) is equal to the index of (Σ,g). In particular, (Σ, g) is L f -stable if and only if (Σ,g) is stable in (M,g).