1 Introduction

The study of sequences of minimal surfaces in 3-dimensional Riemannian manifolds is a classic topic in differential geometry. That kind of compactness is important, for instance, because it leads us to use the Degree Theory in order to study the existence of certain kind of minimal surfaces. The seminal work of Allard [1] deals with weak convergence of sequences of minimal surfaces. However, one of the most celebrated compactness result for closed minimal surfaces in 3-dimensional Riemannian manifolds is the Choi-Schoen’s Compactness Theorem [4] for closed minimal surfaces embedded in a 3-dimensional Riemannian manifold with positive Ricci curvature and bounded genus. Later, B. White generalized the result of Choi and Schoen to stationary points of arbitrary elliptic functionals defined on the space of embeddings of a compact surface in a 3-dimensional Riemannian manifold, minimal surfaces being stationary points of that area functional. White’s result guarantees that compactness holds for such surfaces if we also assume a bound on the area.

As we mentioned above, in 1985, Choi and Schoen [4] proved the following result.

Theorem 1

Let \((M^3,g)\) be a closed (i.e. compact with empty boundary) Riemannian manifold with positive Ricci curvature. Then the space of closed embedded minimal surfaces of fixed topological type in M is compact in the \(C^k\) topology for any \(k\ge 2\).

Motived by this result, in 2015, Li and Wei proved, [9], a generalization of the Choi-Schoen’s Theorem for the class of the f-minimal surfaces.

Theorem 2

Let \((M^{3},g,e^{-f}d\mu )\) be a closed smooth metric measure space with \(Ric_f\ge \kappa >0\). Then the space of closed embedded f-minimal surfaces of fixed topological type in \((M^{3},g,e^{-f}d\mu )\) is compact in the smooth topology.

Li and Wei used the standard argument in Choi–Schoen’s paper. Initially they needed a first eigenvalue estimate of the f-Laplacian for f-minimal surfaces in a manifold with positive Bakry-Émery Ricci curvature \(Ric_f\ge k>0\) (proved by Ma and Du in [10]), then they get an a priori upper bound on the weighted area of the f-minimal surface in terms of the topology. Later they get an upper bound for the total curvature of an f-minimal surface and using a singular compactness result and a contradiction argument they get the smooth compactness Theorem 2.

This paper will prove a similar compactness theorem. Instead \(Ric_f\ge k>0\), we suppose that the smooth metric measure space \((M^{3},g,e^{-f}d\mu )\) contains no immersed f-stable f-minimal surface, that M is simply connected and \(R_f^P+\frac{1}{2}|\overline{\nabla }f|^2>0\), where \(R_f^P\) is the Perelman scalar curvature and \(\overline{\nabla }\) is the Levi-Civita connection of g. Note that \(Ric_f\ge k>0\) implies those two first assumptions but does not imply the latter. In fact, \(R_f^P\) is not the trace of \(Ric_f\). The key idea is to explore the following property: that a surface \(\Sigma \subset M\) is an f-minimal surface in \((M^{3},g,e^{-f}d\mu )\) if and only if \(\Sigma \) is a minimal surface in \((M^3,{\widetilde{g}})\), where \({\widetilde{g}}=e^{-f}g\).

Using results of [1, 3, 12], we get the following theorem.

Theorem 3

Let \((M^{3},g,e^{-f}d\mu )\) be a simply connected and closed smooth metric measure space for which \(R_f^P+\frac{1}{2}|\overline{\nabla }f|^2>0\) where \(R_f^P\) is the Perelman scalar curvature and \(\overline{\nabla }\) is the Levi-Civita connection of g. Assume that \((M^{3},g,e^{-f}d\mu )\) contains no closed f-stable f-minimal surface. Then the space of connected closed embedded f-minimal surfaces of bounded f-index, in \((M^{3},g,e^{-f}d\mu )\), is compact in the \(C^k\) topology for all \(k\ge 2\).

Corollary 1

Let \((M^{3},g,e^{-f}d\mu )\) be a simply connected and closed smooth metric measure space for which the Perelman scalar curvature is positive. Assume that \((M^{3},g,e^{-f}d\mu )\) contains no closed f-stable f-minimal surface. Then the space of connected closed embedded f-minimal surfaces of bounded f-index, in \((M^{3},g,e^{-f}d\mu )\), is compact in the \(C^k\) topology for all \(k\ge 2\).

It is worth pointing out the following facts. Let \((M^{n+1},g,e^{-f}d\mu )\) be a smooth metric measure space with \(Ric_f>0\), then M contains no compact immersed f-stable f-minimal hypersurface (see [8]). If \((M^{3},g,e^{-f}d\mu )\) is a complete smooth metric measure space such that \(Ric_f\ge k>0\), then M has finite weighted volume and finite fundamental group (see [11, 13]); in this case, since \(|\pi _1(M)|<\infty \) after passing to the universal covering, we may assume that M is simply connected. Moreover, if \(Ric_f\ge k>0\); for immersed f-minimal surfaces, uniform bound on their genus imply uniform bound on their f-index (see [3, 6, 9]).

2 Preliminaries

In this section, we will speak a little bit about smooth metric measure spaces, the reader can refer to [2, 7,8,9, 11, 13] and references there in for more information about such spaces. The intention here is only to introduce the definitions of some curvatures and objects that will be used.

By a smooth metric measure space we mean a triple \((M^{n+1},g,e^{-f}d\mu )\), where \((M^{n+1},g)\) is a \((n+1)\)-dimensional smooth Riemannian manifold, f is a smooth function on M and \(d\mu \) is the Riemannian volume measure induced from g. Here, the word closed will mean a space that is compact with empty boundary. For the sake of exposition, we assume here that the hypersurfaces are all two-sided and the ambient manifold is oriented.

The Bakry-Émery Ricci curvature of \((M^{n+1},g,e^{-f}d\mu )\) is defined by

$$\begin{aligned} Ric_f:=Ric+\overline{\nabla }^2f, \end{aligned}$$
(2.1)

where Ric, \(\overline{\nabla }^2\) and \(\overline{\nabla }\) are the Ricci curvature, the Hessian operator and the Levi-Civita connection of g, respectively.

The Perelman’s scalar curvature of \((M^{n+1},g,e^{-f}d\mu )\) is defined by

$$\begin{aligned} R^P_f:=R+2\overline{\Delta }f-\left| \overline{\nabla }f \right| ^2, \end{aligned}$$
(2.2)

where R is the scalar curvature of \((M^{n+1},g)\). Note that \(R^P_f\) is not the trace of \(Ric_f\). In fact, \(tr(Ric_f)=R+\overline{\Delta }f\).

Suppose that \(\Sigma \) is a smooth embedded hypersurface in M, the f-mean curvature of \(\Sigma \) is defined by

$$\begin{aligned} H_f:=H-\left\langle \overline{\nabla }f,\nu \right\rangle , \end{aligned}$$
(2.3)

where \(\nu \) is the unit normal vector field on \(\Sigma \) and H is the mean curvature of \(\Sigma \), in \((M^{n+1},g)\), with respect to \(\nu \). \(\Sigma \) is called an f-minimal hypersurface if its f-mean curvature vanishes everywhere. This definition is motivated by the first variation formula of the weighted volume of \(\Sigma \). The weighted volume of \(\Sigma \) is defined by

$$\begin{aligned} {{\,\mathrm{vol}\,}}_{f}\left( \Sigma \right) :=\int _{\Sigma } e^{-f}dv, \end{aligned}$$
(2.4)

where dv is the volume form of \(\Sigma \) with respect to the metric induced from the ambient space \((M^{n+1},g)\). When \(n=2\), we use \({{\,\mathrm{area}\,}}_f\) instead of \({{\,\mathrm{vol}\,}}_f\). For any normal variation \(\Sigma _s\) of \(\Sigma \) with compactly supported variation vector field \(X=\varphi \nu \) (for some smooth compactly supported function \(\varphi \) on \(\Sigma \)), the first variation formula of \({{\,\mathrm{vol}\,}}_f\) is given by

$$\begin{aligned} \left. \frac{d}{ds}\right| _{s=0}{{\,\mathrm{vol}\,}}_{f} \left( \Sigma _{s}\right) =\int _{\Sigma } \varphi H_f\,e^{-f}dv, \end{aligned}$$
(2.5)

and the second variation is given by

$$\begin{aligned} \left. \frac{d^2}{ds^2}\right| _{s=0}{{\,\mathrm{vol}\,}}_{f} \left( \Sigma _{s}\right) =-\int _{\Sigma } \varphi L_f \varphi \,e^{-f}dv, \end{aligned}$$
(2.6)

where \(L_f:=\Delta _f+|A|^{2}+Ric_f(\nu ,\nu )\), \(\Delta _f:=\Delta +\nabla f\cdot \nabla \), A is the second fundamental form of \(\Sigma \), \(\Delta \) is the Laplacian operator on \(\Sigma \), and \(\nabla \) is the Levi-Civita connection of \((\Sigma ,g)\) induced from \(\overline{\nabla }\). The operator \(\Delta _f\) is called f-Laplacian. More generally, there is the concept of constant weighted mean curvature, see [7].

The f-index of \(\Sigma \), denoted by \(f{{\,\mathrm{\!\mathrm{-}\!}\,}}{{\,\mathrm{index}\,}}(\Sigma )\), is defined to be the dimension of the subspace (of the space of smooth compactly supported functions \(\varphi \) on \(\Sigma \)) such that the following quadratic form is strictly negative definite

$$\begin{aligned} Q^{\Sigma }(\varphi ,\varphi ):=-\int _{\Sigma } \varphi L_f \varphi \,e^{-f}dv, \end{aligned}$$
(2.7)

when \(f{{\,\mathrm{\!\mathrm{-}\!}\,}}{{\,\mathrm{index}\,}}(\Sigma )=0\), we say that \(\Sigma \) is f-stable.

Let \(\Sigma \subset (M^{n+1},g,e^{-f}d\mu )\) be an embedded f-minimal surface. Define a conformal metric \({\widetilde{g}}=e^{-\frac{2}{n}f}g\) on M, denoting by \({{\,\mathrm{vol}\,}}_{{\widetilde{g}}}(\Sigma )\), \(H_{{\widetilde{g}}}(\Sigma )\), \({{\,\mathrm{index}\,}}_{{\widetilde{g}}}(\Sigma )\) and \({\widetilde{R}}\) the volume, the mean curvature, the index of \(\Sigma \subset (M^{n+1},{\widetilde{g}})\), and the scalar curvature of \((M^{n+1},{\widetilde{g}})\) respectively; then \({{\,\mathrm{vol}\,}}_{\widetilde{g}}(\Sigma )={{\,\mathrm{vol}\,}}_f(\Sigma )\), \(H_{\widetilde{g}}(\Sigma )=e^{\frac{f}{n}}H_f(\Sigma )\), \(f{{\,\mathrm{\!\mathrm{-}\!}\,}}{{\,\mathrm{index}\,}}(\Sigma )={{\,\mathrm{index}\,}}_{\widetilde{g}}(\Sigma )\), and \({\widetilde{R}}=e^{\frac{2f}{n}}(R_f^P+\frac{1}{n}|\overline{\nabla }f|^2)\).

3 Proof of the result

Let \(\Sigma _j\subset (M^{3},g,e^{-f}d\mu )\) be a sequence of connected closed embedded f-minimal surfaces such that \(f{{\,\mathrm{\!\mathrm{-}\!}\,}}{{\,\mathrm{index}\,}}(\Sigma _j)\le I\). Define a conformal metric \({\widetilde{g}}=e^{-f}g\) on M, so \({{\,\mathrm{area}\,}}_{\widetilde{g}}(\Sigma _j)={{\,\mathrm{area}\,}}_f(\Sigma _j)\), \(H_{\widetilde{g}}(\Sigma _j)=e^{\frac{f}{2}}H_f(\Sigma _j)\), \(f{{\,\mathrm{\!\mathrm{-}\!}\,}}{{\,\mathrm{index}\,}}(\Sigma _j)={{\,\mathrm{index}\,}}_{\widetilde{g}}(\Sigma _j)\) and \({\widetilde{R}}=e^f(R_f^P+\frac{1}{2}|\overline{\nabla }f|^2)\). Since \(H_{\widetilde{g}}(\Sigma _j)=0\), then \(\Sigma _j\) is also a sequence of connected, closed, embedded minimal surfaces in \((M^3,{\widetilde{g}})\). As \(R_f^P+\frac{1}{2}|\overline{\nabla }f|^2>0\), we have \({\widetilde{R}}>0\). From [3, Theorem 1.3]; there is \(A_0=A_0(M,{\widetilde{g}},I)<\infty \) and \(r_0=r_0(M,{\widetilde{g}},I)\) so that \({{\,\mathrm{area}\,}}_{\widetilde{g}}(\Sigma _j)\le A_0\) and \({{\,\mathrm{genus}\,}}(\Sigma _j)\le r_0\). Therefore, \({{\,\mathrm{area}\,}}_{\widetilde{g}}(\Sigma _j)\le A_0\) and \({{\,\mathrm{index}\,}}_{\widetilde{g}}(\Sigma _j)\le I\). From [12, Theorem 2.3] we obtain a closed connected and embedded minimal surface \(\Sigma \subset (M,{\widetilde{g}})\) where \(\Sigma _j{\mathop {\longrightarrow }\limits ^{{\scriptscriptstyle (M,{\widetilde{g}})}}}\Sigma \) in the varifold sense and \({{\,\mathrm{index}\,}}_{\widetilde{g}}(\Sigma )\le I\). Moreover, assuming that \(\Sigma _j\not =\Sigma \) eventually, then the convergence is smooth and graphical for all \(x\in M\backslash \mathcal B\), where \(\mathcal B\subset M\) is a finite set of points with \(|\mathcal B|\le I\).

We have to show that the convergence holds across the points of \(\mathcal B\), i.e., the convergence is smooth everywhere. By Allard’s regularity theorem (see [1]), this follows from showing that the convergence is of multiplicity one.

We know that 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 will be stable (see [5]). Therefore, if \(\Sigma _j\rightarrow \Sigma \) in \((M,{\widetilde{g}})\) with multiplicity greater than one, then the limit minimal surface \(\Sigma \subset (M,{\widetilde{g}})\) will be stable, therefore \(\Sigma \subset (M^{3},g,e^{-f}d\mu )\) will be f-stable, this contradicts the hypothesis.