2.1. Morse index of $f$-minimal hypersurfaces with free boundary

In this subsection, we establish the notion of the Morse index in the setting of two-sided $f$-minimal hypersurfaces with free boundary; for more details, refer to [Reference Castro and Rosales7].

A hypersurface $M^n$ in $(\overline {M},\,g,\,{\rm e}^{-f} {\rm d}\mu )$ with boundary $\partial M \subset \partial \overline {M}$ is considered a free-boundary hypersurface if $M$ intersects $\partial \overline {M}$ orthogonally. In other words, if $\eta$ denotes the unitary conormal vector field of $\partial M$ at $\overline {M}$, pointing outwards, then $M$ is considered a free-boundary hypersurface when $\eta$ is orthogonal to $T(\partial \overline {M})$.

Given a normal variation $M_t$ associated with the variational field $uN$, $u\in C^\infty$, the formula for the first variation of the $f$-volume is

\[ \frac{{\rm d}}{{\rm d}t}\bigg|_{t=0} {\rm{Vol}}_f (M_t)={-}\int_M uH_f {\rm e}^{{-}f}{\rm d}\mu + \int_{\partial M} ug(\eta,N) {\rm e}^{{-}f}{\rm d}\sigma, \]

where $H_f = H + \langle N,\overline {\nabla } f\rangle$ is the $f$-mean curvature of $M$ in $\overline {M}$. Therefore, $M$ is critical (or stationary) for the $f$-volume if, and only if, $M$ is $f$-minimal with free boundary. Let $M$ be a $f$-minimal hypersurface with free boundary, the quadratic form associated with the second variation of the $f$-volume of $M$ in the direction of the normal field $uN$ is given by

\begin{align*} Q_f (u,u) & := \frac{{\rm d}^2}{{\rm d}t^2}\bigg|_{t=0} {{\rm{Vol}}}_f(M)\\ & ={-}\int_M u L_f u {\rm e}^{{-}f}{\rm d}\mu + \int_{\partial M} u \left( \eta(u) + h^{\partial \overline{M}}(N,N) u \right) {\rm e}^{{-}f}{\rm d}\sigma\\ & = \int_M \left( |\nabla u|^2 - ({\rm{Ric}}_f(N,N)+|A|^2)u^2 \right){\rm e}^{{-}f}{\rm d}\mu + \int_{\partial M} u^2 h^{\partial \overline{M}}(N,N) {\rm e}^{{-}f}{\rm d}\sigma, \end{align*}

where $h^{\partial \overline {M}}(N,\,N)=-\langle \bar \nabla _N\nu,\,N\rangle$ is the second fundamental form of $\partial \overline {M}$ in $\overline {M}$ with respect to the outward normal vector field $\nu$, ${\rm {Ric}}_f = {\rm {Ric}} + {\rm Hess} f$, $|A|$ is the norm of the second fundamental form of $M$, and $L_f = \Delta _f+{\rm {Ric}}_f(N,\,N)+|A|^2$ is the Jacobi operator of the immersion. We say that $\lambda (L_f)$ is an eigenvalue of $L_f$ with eigenfunction $u\in C^\infty (M)$ if

\[ \begin{cases} L_f u + \lambda u=0 & \text{ on } M, \\ \eta(u) + h^{\partial \overline{M}}(N,N) u = 0 & \text{ in } \partial M. \end{cases} \]

Notice that the boundary condition makes the Jacobi operator self-adjoint. Therefore, it follows from the classical partial differential equation theory that there is a non-decreasing sequence of eigenvalues $\{\lambda _k \}_{k=1}^\infty$ associated with an orthonormal basis $\{u_k \}_{k=1}^\infty$ of $L ^2(M,\,{\rm e}^{-f}{\rm d}\mu )$. Recall the Morse index of $M$, ${\rm {Ind}}_f(M)$, is given by the number of negative eigenvalues of $L_f$ counting with multiplicity.

2.2. $f$-Harmonic 1-forms on manifolds with boundary

Throughout this paper, we denote the inclusion mapping of $\partial M$ into $M$ by $i$, and $i^\ast$ denotes its pullback; $d$ is the outer derivative operator; and $\delta :=(-1)^{n(p+1)+1}\star d \star$ is the inner derivative operator, where $\star :\Omega ^p (M)\to \Omega ^{n-p}(M)$ is the Hodge star operator. In our setting, we have the weighted interior derivative operator defined by $\delta _f=\delta +\iota _{\nabla f}$, where $\iota _X$ is the contraction operator from the left hand by $X$, see [Reference Impera, Rimoldi and Savo11]. Finally, we have the Hodge $f$-Laplacian acting on $p$-forms that is denoted by $\Delta _f^{[p]}$, and defined naturally as

\[ \Delta_f^{[p]} := {d}\delta_f + \delta_f d. \]

Following the standard notation in literature, a $p$-form $\omega$ is called $f$-harmonic if ${ d}\omega =0$ and $\delta _f \omega =0$. We note that on manifolds with boundary the set of $p$-forms that $\Delta _f^{[p]} \omega = 0$ may be different from the set of $f$-harmonic $p$-forms.

Regarding the behaviour of a $p$-form on the boundary, we say that a $p$-form $\omega$ is normal on the boundary whether $i^\ast \omega =0$ or, equivalently, if $\eta \wedge \omega =0$ on $\partial M$. Furthermore, $\omega$ is said to be tangential on the boundary whether $i^\ast (\star \omega )=0$, that is, if $\iota _\eta \omega =0$ on $\partial M$. We denote the spaces of the tangent and normal $f$-harmonic $p$-forms on the boundary, respectively, by

\begin{align*} H_{Nf}^p (M) & = \{ \omega \in \Omega^p(M) : { d}\omega=0, \delta_f \omega=0 \text{ and } \iota_\eta \omega=0 \text{ on } \partial M \},\\ H_{Tf}^p (M) & = \{ \omega \in \Omega^p(M) : { d}\omega=0, \delta_f \omega=0 \text{ and } i^\ast \omega=0 \text{ on } \partial M \}. \end{align*}

A $p$-form $\omega$ satisfies the relative boundary condition if both $\omega$ and $\delta _f \omega$ are normal on the boundary. If $\omega$ and ${ d}\omega$ are tangential on the boundary, we say that $\omega$ satisfies the absolute boundary condition. For the case where $f=0$, refer to [Reference Ambrozio, Carlotto and Sharp5].

Lemma 2.1 Let $(M,g,{\rm e}^{-f}{\rm d}\mu )$ be a compact with possible non-empty boundary smooth metric measure space. Given $\omega \in \Omega ^p(M)$, we have

\[ \int_M \left(\langle \Delta_f^{[p]} \omega, \omega \rangle - |{ d}\omega|^2 - |\delta_f \omega|^2 \right) {\rm e}^{{-}f}{\rm d}\mu = \int_{\partial M} \left( \langle i^\ast \delta_f \omega, \iota_\eta \omega \rangle - \langle i^\ast \omega, \iota_\eta {\rm d}\omega \rangle \right) {\rm e}^{{-}f} {\rm d}\sigma. \]

Proof. Consider the forms $\alpha \in \Omega ^p(M)$ and $\beta \in \Omega ^{p+1}(M)$. From [Reference Yano17, Chapter 8], we have

\[ \langle d({\rm e}^{{-}f}\alpha),\beta \rangle - \langle {\rm e}^{{-}f}\alpha, \delta \beta \rangle ={-}\delta(\beta\llcorner ({\rm e}^{{-}f}\alpha)), \]

and using integration by parts, we get:

\[ \int_M \langle d({\rm e}^{{-}f}\alpha),\beta \rangle - \langle {\rm e}^{{-}f}\alpha, \delta \beta \rangle {\rm d}\mu = \int_{\partial M} \langle {\rm e}^{{-}f} i^\ast \alpha, \iota_\eta \beta \rangle {\rm d}\sigma = \int_{\partial M} \langle i^\ast \alpha, \iota_\eta \beta \rangle {\rm e}^{{-}f}{\rm d}\sigma. \]

On the contrary, a direct computation yields that

\begin{align*} \int_M \langle d({\rm e}^{{-}f}\alpha),\beta \rangle - \langle {\rm e}^{{-}f}\alpha, \delta \beta \rangle {\rm d}\mu & = \int_M \langle {\rm e}^{{-}f}{ d}\alpha,\beta \rangle + \langle \alpha \wedge {\rm d}({\rm e}^{{-}f}),\beta\rangle - \langle {\rm e}^{{-}f}\alpha, \delta \beta \rangle {\rm d}\mu\\ & = \int_M \langle {\rm e}^{{-}f}{ d}\alpha,\beta \rangle - \langle {\rm e}^{{-}f}\alpha,\iota_{\nabla f}\beta\rangle - \langle {\rm e}^{{-}f}\alpha, \delta \beta \rangle {\rm d}\mu\\ & = \int_M \left( \langle { d}\alpha,\beta \rangle - \langle \alpha,\delta_f \beta \rangle \right) {\rm e}^{{-}f}{\rm d}\mu. \end{align*}

Combining the last two equalities, we obtain that

\[ \int_M \left( \langle { d}\alpha,\beta \rangle - \langle \alpha,\delta_f \beta \rangle \right) {\rm e}^{{-}f}{\rm d}\mu= \int_{\partial M} \langle i^\ast \alpha, \iota_\eta \beta \rangle {\rm e}^{{-}f}{\rm d}\sigma. \]

Therefore,

\begin{align*} \int_M \langle \Delta_f^{[p]} \omega, \omega \rangle {\rm e}^{{-}f}{\rm d}\mu & = \int_M \langle \delta_f { d}\omega, \omega \rangle {\rm e}^{{-}f}{\rm d}\mu + \int_M \langle { d}\delta_f \omega, \omega \rangle {\rm e}^{{-}f}{\rm d}\mu\\ & = \int_M |{ d}\omega|^2 {\rm e}^{{-}f}{\rm d}\mu - \int_{\partial M} \langle i^\ast \omega, \iota_\eta { d}\omega \rangle {\rm e}^{{-}f}{\rm d}\sigma \\ & \quad+ \int_M |\delta_f \omega|^2 {\rm e}^{{-}f}{\rm d}\mu + \int_{\partial M} \langle i^\ast \delta_f \omega, \iota_\eta \omega \rangle {\rm e}^{{-}f}{\rm d}\sigma,\\ \end{align*}

for any $\omega \in \Omega ^p(M)$ as desired.

As a direct consequence of lemma 2.1, we can also characterize the spaces $H_{Nf}^p(M)$ and $H_{Tf}^p(M)$ respectively as follows:

\begin{align*} H_{Nf}^p (M) & \!= \{ \omega \!\in\! \Omega^p(M) : \Delta_f^{[p]} \omega=\!0\ \text{and } \omega \text{ satisfies the absolute boundary condition} \},\\ H_{Tf}^p (M) & \!= \{ \omega \!\in\! \Omega^p(M) : \Delta_f^{[p]} \omega=\!0 \text{ and } \omega \text{ satisfies the relative boundary condition} \}. \end{align*}

From the Hodge decomposition, it is well known that $H_N^p(M)\cong H^p(M; \mathbb {R})$. Furthermore, the Hodge star operator $\star$ induces an isomorphism between $H_N^p(M)$ and $H_T^{n-p}(M)$. In general, the following isomorphisms hold:

\[ H_T^p(M)\cong H_N^{n-p}(M)\cong H^{n-p}(M;\mathbb{R})\cong H_p(M,\partial M;\mathbb{R}) . \]

An important fact is that the Hodge decomposition is still valid on smooth metric measure spaces, see [Reference Bueler6]. For the case of 1-forms, we have the following isomorphisms:

• • $H_N^1(M)\cong H_{Nf}^1(M)$ via isomorphism $\omega \mapsto \omega + {\rm d}u$ for $\omega \in H_N^1(M)$, where $u\in C^\infty (M)$ is a solution to the boundary problem:

  \[ \begin{cases} \Delta_f u ={-}\iota_{\nabla f}\omega & \text{ on } M, \\ \displaystyle\frac{\partial u}{\partial \eta}=0 & \text{ in } \partial M. \end{cases} \]

• • $H_T^1(M)\cong H_{Tf}^1(M)$ via isomorphism $\omega \mapsto \omega + {\rm d}u$ for $\omega \in H_T^1(M)$, where $u\in C^\infty (M)$ is a solution to the boundary problem:

  \[ \begin{cases} \Delta_f u ={-}\iota_{\nabla f}\omega & \text{ on } M, \\ \displaystyle\frac{\partial u}{\partial \eta}={-}\iota_\eta \omega, u=0 & \text{ in } \partial M. \end{cases} \]

In particular, the dimension of $H_{Nf}^1(M)$ is equal to $\dim H^1(M; \mathbb {R})$. Throughout the paper, we use the isomorphisms $H_{Nf}^1(M)\cong H^1(M;\mathbb {R})$ and $H_{Tf}^1(M)\cong H^{n-1}(M;\mathbb {R})$.

2.3. Singular manifolds

In this subsection, we will set up some terminology inspired by the discussion found in [Reference Zhou18, Sections 1 and 2]. Let $\overline {M}^{n+1}$ be an $(n+1)$-dimensional connected, compact, orientable Riemannian manifold and $M\subset \overline {M}$ a closed subset. The regular part of $M$ is defined as

\[ {\rm reg}(M) := \{x\in M : M \text{ is a smooth embedded hypersurface near } x \}, \]

and the singular part is ${\rm sing}(M) := M\setminus {\rm reg}(M)$. Clearly, the regular part ${\rm reg}(M)$ is an open subset of $M$.

By a singular hypersurface with a singular set of Hausdorff codimension no less than $k$, $k \in \mathbb {N}$ and $k < n$, we mean a closed subset $M$ of $\overline {M}$ with finite $n$-dimensional Hausdorff measure $\mathcal {H}^n(M) < \infty$ and the $(n - k+\varepsilon )$-dimensional Hausdorff measure $\mathcal {H}^{n-k+\varepsilon }({\rm sing}(M)) = 0$, for all $\varepsilon > 0$. Later on, we will denote $M = {\rm reg}(M)$ and also call $M$ a singular hypersurface; see [Reference Zhou18] and references therein for more details.

To study the Morse index on hypersurfaces with singularities, we must use test functions that allow us to deal with singularities in the moment of integrate. For this purpose, it will be necessary to present the cut-off functions given in the following proposition, which was originally created by Morgan and Ritoré in [Reference Morgan and Ritoré12], and reproduced by Zhu in [Reference Zhu19].

5.1. Free-boundary self-shrinkers in the half-space $\Omega =\mathbb {R}^{n+1}_+$

Consider the Euclidean half-space given by $\Omega = \{(x_1,\,\ldots,\,x_{n+1})\in \mathbb {R}^{n+1}: x_{n+1}\geq 0\}.$ Notice that $\partial \Omega = \mathbb {R}^n\times \{0\}$, and so its unit outward normal vector field is $\nu =-e_{n+1}$. In particular, the $f$-mean curvature of $\partial \Omega$ is equal to zero, and thus $\partial \Omega$ is $f$-mean convex.

Let $M$ be a compact, orientable, free-boundary self-shrinker of mean curvature flow in $\Omega$. Observe that a straightforward computation gives us that $\Delta _fx_i+x_i=0$ and ${\partial x_i}/{\partial \nu }=0$, for $n=1,\ldots,n$. Hence, $\lambda =1$ is eigenvalue of the $f$-Laplacian less than $2\alpha =2$ with multiplicity $n$. Thus, we obtain:

Theorem 5.1 Let $M^n$ be a free-boundary self-shrinker in the half-space $\mathbb {R}^{n+1}_+$. Then,

\[ {\rm{Ind}}_{{|x|^2}/{2}}(M)\geq \frac{2}{n(n+1)} \left(\dim H^{n-1} (M;\mathbb{R}) + n \right). \]

Another strategy is consider the function $\langle e_{n+1},N\rangle$. It is well known that the functions $\left \langle \overline {V},\,N \right \rangle$ satisfy the equation:

\[ L_f \left\langle \overline{V},N \right\rangle -\left\langle \overline{V},N \right\rangle=0, \quad \forall \overline{V} \in \overline{\textbf{P}} . \]

So, $\langle \overline {V},\,N \rangle$ will be an eigenfunction of $L_f$ associated with the eigenvalue $\lambda =-1 \in (-2,0)$, if:

\[ \eta (\left\langle \overline{V},N \right\rangle)+ h^{\partial\Omega}(N,N)\left\langle \overline{V},N \right\rangle=0, \]

that is:

\[ 0=\eta(\langle \overline{V},N \rangle)={-}\langle \overline{V}, A^M\eta\rangle \text{ on } \partial M. \]

Since the hyperplane is totally geodesic and the hypersurface $M$ is free boundary, a direct computation shows that $A^M\eta = k\eta$, for some function $k$ along $\partial M$, and therefore the function $\langle \overline {V},\,N \rangle$ satisfies the Neumann condition for any $\overline V$ orthogonal to $\eta =-e_{n+1}$.

Hence, the space $Z:= span \{ \overline {V}\in \overline {\textbf {P}}: \eta \langle \overline {V},\,N \rangle =0$ on $\partial M\}$ has dimension at least $n$, and so:

\[ \Gamma_{L_f}^-({-}2)\geq \dim Z \geq n. \]

Theorem 5.2 Let $M^n$ be a free-boundary self-shrinker in the half-space $\mathbb {R}^{n+1}_+$. Then,

\[ {\rm{Ind}}_{{|x|^2}/{2}}(M)\geq \frac{2}{n(n+1)} \dim H^{n-1} (M;\mathbb{R}) +n. \]

In particular, for $n=2$, we have

\[ {\rm{Ind}}_{{|x|^2}/{2}}(M)\geq \frac{1}{3}(2g+r+5). \]

5.2. Free-boundary self-shrinkers into a slab

Using the previous strategy we obtain:

Theorem 5.4 Let $M^2$ be a free-boundary self-shrinker in a slab $\mathbb {R}^2\times [a,b]$, where $a< b$ are real numbers. Then,

\[ {\rm{Ind}}_{{|x|^2}/{2}}(M)\geq \dfrac{2g+r+5}{3}. \]

Notice that the Morse index equal to two implies a topological rigidity. Indeed: