Low index capillary minimal surfaces
in Riemannian $3$-manifolds

Minimal surfaces are critical points for the area functional under suitable constraints. If we only consider closed surfaces, no constraints are necessary. If the surfaces have a fixed boundary, this leads to the so called Plateau problem, first studied by Lagrange [11] and Meusnier [14]. However, there is a third situation that can be investigated: when we consider compact surfaces whose boundaries are allowed to move freely in the boundary of the ambient manifold. Not surprisingly, critical points of the area functional under this constraint are called free boundary minimal surfaces. The boundary of such surfaces meet the boundary of the ambient manifold at a $90^{\circ}$ angle.

Although the first works dealing with this type of surfaces date back to 1938 with R. Courant (see [8] and [9]), in the last decade there have been incredible developments in this field, with the employment of new techniques and the emergence of interesting conceptual links. Among the main contributors to this topic of research we could cite Fraser, Chen and Pang, with their work on free boundary surfaces on positively curved ambients [6], Ambrozio with his work on rigidity of mean-convex manifolds [2], and additionally Carlotto and Sharp (jointly with Ambrozio), with their works on compactness analysis and index estimates for free boundary minimal hypersurfaces ([3] and [4]).

In this paper we are interested in a natural generalisation of free boundary minimal surfaces, namely, capillary minimal surfaces. These are critical points of a certain energy functional, which will be presented in Section 2. As will be deduced later, they can be characterised as minimal surfaces whose boundary meet the ambient boundary at a constant angle — the capillary angle.

Capillary surfaces in $\mathbb{R}^{3}$ model the configuration of liquids in containers in the absence of gravity. In fact, the interface between the fluid and the air is a surface with boundary that (locally) minimises the energy functional. This energy depends on the area of the interface, the area wetted by the fluid in the container and the angle of contact between the surface and the boundary of the container. More general situations have also been considered in the literature, like the influence of gravity and the density of the fluid in the equilibrium shape. We refer the interested reader to the book of Finn [10] for an extensive survey on this subject and the derivation of the equations that describe such surfaces.

Like in the free boundary case, questions relating the topology and the geometry of surfaces raise a lot of attention from geometers. For instance, given an ambient manifold $(M^{3},g)$ of a particular shape, what are the possible topological types of surfaces $\Sigma$ that admit a capillary CMC or minimal embedding into $M$? Is it possible to characterise the geometry of the allowed types? The first result in this direction was obtained by Nitsche [15], who proved that any immersed capillary disc in the unit ball of $\mathbb{R}^{3}$ must be either a spherical cap or a flat disc. Later, Ros and Souam [18] extended this result to capillary discs in balls of $3$-dimensional space forms. Recently, Wang and Xia [21] analysed the problem in an arbitrary dimension and proved that any stable immersed capillary hypersurface in a ball in space forms is totally umbilical. There are many interesting uniqueness results in other types of domains, like slabs [1], wedges [16, 7], cylinders [13] and cones [17].

We are interested in the same questions raised above, but in a more general setting. Namely, we only impose curvature assumptions on the ambient $3$-manifold and look for restrictions in the topology of the possible immersed (or embedded) capillary minimal surfaces.

When dealing with the free boundary case, Ambrozio introduced the following functional in the space of compact and properly immersed surfaces in a Riemannian $3$-manifold $M$:

where $R_{M}$ is the scalar curvature of $M$, $H_{\partial M}$ is the mean curvature of $\partial M$, $|\Sigma|$ denotes the area of $\Sigma$ and $|\partial\Sigma|$ denotes the length of $\partial\Sigma$.

As a first result, we modify Ambrozio’s functional to take care of the capillary case and we show the following theorem, generalising Proposition 6 in [2]:

A compact two-sided capillary minimal surface, properly embedded in $(M,g)$ with contact angle $\theta\in(0,\pi)$, that satisfies conditions (i), (ii) and (iii) of Theorem A will be called infinitesimally rigid. Given one such surface $\Sigma$, there is a way to obtain a vector field $Z$ in $\Sigma$ such that $g(Z,N)=1$ and $Z$ is tangent to $\partial M$ along $\partial\Sigma$. Let us also denote by $Z$ an extension to $M$ which is tangent to $\partial M$ along the entire boundary of $M$. Let $\phi=\phi(x,t)$ the local flow of $Z$ and fix a number $\alpha$ between $0$ and $1$. We show the existence of a local foliation around $\Sigma$, employing the same techniques as Ambrozio in Proposition 10 in [2]:

It is also possible to show the existence of a capillary minimal foliation, where the contact angles now vary from leaf to leaf. Although it will not be used subsequently, we believe it may be of independent interest (see Remark 3.4).

Then, we use Theorem B to show that, under some hypotheses, a dichotomy occurs: either the contact angle is equal to $\pi/2$ or a very special situation takes place. More precisely, we have:

A situation when $\theta\neq\pi/2$ may happen in Theorem D is the following. Let $P_{1}$ and $P_{2}$ be two non parallel planes in $\mathbb{R}^{3}$, intersecting along a line $\ell$, and let $Q$ be a plane in $\mathbb{R}^{3}$ which is parallel to $\ell$ and intersects both $P_{1}$ and $P_{2}$ at the same angle. Fix $S_{0}$ to be the (closed) wedge determined by $P_{1}$ and $P_{2}$ which contains both $Q\cap P_{1}$ and $Q\cap P_{2}$, and let $S=S_{0}\setminus\ell$ (see Figure 1). Now fix $T$ a translation of $\mathbb{R}^{3}$ by a vector parallel to the line $\ell$ and let $M$ be the quotient of $S$ by the group $G$ generated by $T$. If we define $\Sigma$ to be the quotient of $Q\cap S$ by $G$, then $\Sigma$ is an infinitesimally rigid cylinder intersecting $\partial M$ at a constant angle, (a) and (b) hold, $M$ is flat and $\partial M$ is totally geodesic, as we wanted. One question remains: is $\Sigma$ energy-minimising? We believe so, but we did not find a proof.

Next, we prove two other inequalities relating the geometry and the topology of capillary minimal surfaces of low index. This result generalises Theorem 1.2 in [6] to the capillary case. We note that item (i) below also generalises the free boundary case.

The author would like to thank Paolo Piccione for his constant support and encouragement during the period when this article was written and revised, and for countless fruitful mathematical conversations. He also expresses sincere gratitude to Lucas Ambrozio for valuable comments on this work. Additionally, the author thanks Izabella Freitas and Jackeline Conrado for the figures in this paper.

The purpose of this section is to formally introduce the concept of capillary CMC and minimal hypersurfaces. Despite the fact that in section 3 we will be dealing only with capillary surfaces in Riemannian $3$-manifolds, there is no significant simplification in introducing the main concepts only in dimension $2$. So, the general situation will be addresed in the sequel.

Let $(M^{n+1},g)$ be a Riemannian manifold with nonempty boundary. Let $\Sigma^{n}$ be a smooth compact manifold with nonempty boundary, and let $\varphi:\Sigma\to M$ be a smooth immersion of $\Sigma$ into $M$. We say that $\varphi$ is a proper immersion if $\varphi(\Sigma)\cap\partial M=\varphi(\partial\Sigma)$.

Henceforth, we assume that $\varphi$ is two-sided. Fix a unit normal vector field $N$ for $\Sigma$ along $\varphi$ and denote by $\nu$ the outward unit conormal for $\partial\Sigma$ in $\Sigma$. Moreover, let $\overline{N}$ the outward pointing unit normal for $\partial M$ and let $\overline{\nu}$ the unit normal for $\partial\Sigma$ in $\partial M$ such that the bases $\{N,\nu\}$ and $\{\overline{N},\overline{\nu}\}$ determine the same orientation in $(T\partial\Sigma)^{\perp}$. See Figure 2 to gain some intuition.

A smooth function $\Phi:\Sigma\times(-\varepsilon,\varepsilon)\to M$ is called a proper variation of $\varphi$ is the maps $\varphi_{t}:\Sigma\to M$, defined by $\varphi_{t}(x)=\Phi(x,t)$, are proper immersions for all $t\in(-\varepsilon,\varepsilon)$, and if $\varphi_{0}=\varphi$.

Let us fix a proper variation $\Phi$ of $\varphi$. The variational vector field associated to $\Phi$ is the vector field $\xi_{\Phi}=\xi:\Sigma\to TM$ along $\varphi$ defined by

We now define some important functionals related to the variation $\Phi$. The area functional $A:(-\varepsilon,\varepsilon)\to\mathbb{R}$ is given by

where $\mathrm{d}A_{\varphi_{t}^{*}g}$ denotes the area element of $(\Sigma,\varphi_{t}^{\ast}g)$. Even if $n=\dim\Sigma>2$, it is customary to refer to this as the area functional.

The volume functional $V:(-\varepsilon,\varepsilon)\to\mathbb{R}$ is defined by

where $\mathrm{d}V$ is the volume element of $M$. We say that the variation $\Phi$ is volume preserving if $V(t)=0$ for every $t\in(-\varepsilon,\varepsilon)$.

We also consider the wetting area functional $W:(-\varepsilon,\varepsilon)\to\mathbb{R}$:

where $\mathrm{d}A_{\partial M}$ denotes the area element of $\partial M$.

Finally, we define the energy functional. In order to do so, let us fix an angle $\theta\in(0,\pi)$. Then $E_{\Phi,\theta}=E:(-\varepsilon,\varepsilon)\to\mathbb{R}$ is given by

The following proposition contains the formulae for the first variation of the energy and volume, whose proof can be found in [20] and [5]:

We say that the immersion $\varphi$ is a capillary CMC immersion if $E^{\prime}(0)=0$ for every volume preserving variation of $\varphi$. If $E^{\prime}(0)=0$ for every variation of $\varphi$, we call $\varphi$ a capillary minimal immersion. When there is only one immersion under consideration and there is no risk of confusion, we just say that $\Sigma$ is a capillary CMC or minimal hypersurface. See Figure 2 for an example in an Euclidean cone.

Notice that $\Sigma$ is a capillary CMC hypersurface if and only if $\Sigma$ has constant mean curvature and $g(N,\overline{N})=\cos\theta$ along $\partial\Sigma$; this last condition means that $\partial\Sigma$ meets $\partial M$ at an angle of $\theta$. Similarly, $\Sigma$ is a capillary minimal hypersurface when $\Sigma$ is a minimal hypersurface and $\partial\Sigma$ meets $\partial M$ at an angle of $\theta$. When $\theta=\pi/2$, we use the term free boundary CMC (or minimal) hypersurface.

For a capillary CMC or minimal hypersuface $\Sigma$ with contact angle $\theta\in(0,\pi)$, the orthonormal bases $\{N,\nu\}$ and $\{\overline{N},\overline{\nu}\}$ are related by the following equations:

This will be important in the calculations.

We now discuss the notion of stability. Firstly, let us fix some notation. For a proper immersion $\varphi:\Sigma\to M$, let $A(X,Y)=g(-\nabla_{X}N,Y)$ the second fundamental form of $\Sigma$, and denote by $\Delta_{\Sigma}$ the Laplacian of $\Sigma$ with respect to the metric induced by $\varphi$. Moreover, let $\operatorname{\mathrm{I\!I}}(v,w)=g(\nabla_{v}\overline{N},w)$ be the second fundamental form of $\partial M$ with respect to $-\overline{N}$.

The following proposition gives a formula for the second variation of energy. The proof is long and can be found in the Appendix of [18].

The capillary CMC immersion $\varphi:\Sigma\to M$ (or just $\Sigma$) is called stable if $E^{\prime\prime}(0)\geq 0$ for any volume preserving variation of $\varphi$. If $\varphi$ is a capillary minimal immersion, we call it stable whenever $E^{\prime\prime}(0)\geq 0$ for every variation of $\varphi$.

Alternatively, let $\mathcal{F}=\{f\in H^{1}(\Sigma):\int_{\Sigma}f\,\mathrm{d}A=0\}$, where $H^{1}(\Sigma)$ is the first Sobolev space of $\Sigma$. The index form $Q:H^{1}(\Sigma)\times H^{1}(\Sigma)\to\mathbb{R}$ of $\Sigma$ is given by

Then $\varphi$ is a capillary CMC stable immersion if and only if $Q(f,f)\geq 0$ for every $f\in\mathcal{F}$. If $\varphi$ is a capillary minimal immersion, then it is stable precisely when $Q(f,f)\geq 0$ for every $f\in H^{1}(\Sigma)$.

The stability index of a capillary CMC (resp. minimal) hypersurface $\Sigma$ is the dimension of the largest vector subspace of $\mathcal{F}$ (resp. $H^{1}(\Sigma)$) restricted to which the bilinear form $Q$ is negative definite. Thus, stable hyperfurfaces are those which have index equal to zero. Intuitively, this means that no variation of a capillary minimal stable hypersurface decreases area up to second order.

It is possible to analitically compute the stability index of a capillary CMC (resp. minimal) hypersurface $\Sigma$ in terms of the spectrum of its Jacobi operator with a Robin boundary condition. Indeed, the boundary condition

is elliptic for the Jacobi operator $L_{\Sigma}$. Therefore, there exists a nondecreasing and divergent sequence $\lambda_{1}\leq\lambda_{2}\leq\cdots\leq\lambda_{k}\nearrow\infty$ of eigenvalues for the problem

The index of $\Sigma$ is then the number of negative eigenvalues of the problem above associated to (smooth) eigenfunctions that lie in $\mathcal{F}$ (resp. $H^{1}(\Sigma)$) (see more details in [6] and [19]).

This section is devoted to prove the Theorems mentioned in the Introduction.

Recall that, given a Riemannian $3$-manifold $(M^{3},g)$ with nonempty boundary and a fixed angle $\theta\in(0,\pi)$, we can define the following functional in the space of compact properly immeresd surfaces in $M$:

provided that the scalar curvature $R_{M}$ and the mean curvature $H_{\partial M}$ are bounded from below.

We first show that if $\Sigma$ is capillary minimal stable, then $\operatorname{I}_{\theta}(\Sigma)$ can be bounded above by the topology of $\Sigma$. We also characterise the equality case.

We begin with a simple lemma.

We are amost ready to prove Theorem B. Recall that $Z$ is a vector field in $M$ which is tangent to $\partial M$ along the entire boundary of $M$ and which satisfies $g(Z,N)=1$ along $\Sigma$. Let $\varphi=\varphi(x,t)$ be the local flow of $Z$ and fix $\alpha\in(0,1)$.

We begin with a lemma, whose proof can be found in the appendices of [2] and [12].