We consider a variation of the submanifold \(\Sigma\) given by a family of immersions \(\iota_t: \Sigma \to M\) with \(\iota_0 = \iota\). The variation vector field is defined as
\[
V = \frac{\partial \iota_t}{\partial t}\bigg|_{t=0}.
\]
The volume element of \(\Sigma_t = \iota_t(\Sigma)\) is denoted by \(d\Sigma_t\).
Suppose \(U \subset \Sigma\) is an open subset. We consider the variation of the volume functional
\[
\mathcal{A}(t)=|\iota_t(U)| = \int_{U} d\Sigma_t.
\]
Proposition 1.3.1
The first variation of the volume functional is given by
\[
\delta_V(U):=\mathcal{A}'(0) = \int_{ \Sigma}\mathrm{div}^\Sigma V \, d\Sigma.
\]
The second variation of the volume functional is given by
\[
\begin{aligned}
\delta^{2}_V(U):={}\mathcal{A}''(0) = \int_\Sigma \sum_{i=1}^n-\left< R(V,e_i)V,e_i \right> + \mathrm{div}^\Sigma \nabla_{V}V + (\mathrm{div}^\Sigma V)^{2} \\
+ \sum_{i=1}^{n}|(\nabla_{e_i}V)^\bot|^{2} - \sum_{i,j=1}^n \left< \nabla_{e_i}V, e_j\right> \left< \nabla_{e_j}V, e_i \right> \, d\Sigma.
\end{aligned}
\]
Note that we do not impose any conditions on \(V\) and \(\Sigma\) for the above formulas. In particular, \(\Sigma\) does not need to be a critical point of the volume functional, and \(V\) does not need to be a normal variation.
Suppose \(\{e_i\}\) is an orthonormal basis of \(T\Sigma\), and define \(e_i(t)=\iota_{t*} e_i\). Then the area element can be expressed as
\[
d\Sigma_t = \sqrt{\det(\left< e_i(t),e_j(t) \right>)} \, d\Sigma,
\]
where \(d\Sigma\) is the area element of \(\Sigma\). We need the following lemma:
Lemma 1.3.2
We have
\[
\mathrm{det}(I+tA) = 1 + t \cdot \mathrm{tr}(A) + \frac{t^{2}}{2}(\mathrm{tr}(A)^{2} - \mathrm{tr}(A^{2})) + O(t^{3}),
\]
Proof. We recall the standard formula for the determinant of a matrix \(M\):
\[
\mathrm{det}(M) = \exp(\mathrm{tr}(\log M)).
\]
Applying this to \(I+tA\), we have
\[
\begin{aligned}
\mathrm{det}(I+tA) &= \exp(\mathrm{tr}(\log(I+tA))) \\
&= \exp\left(\mathrm{tr}\left(tA - \frac{t^{2}}{2}A^{2} + O(t^{3})\right)\right) \\
&= \exp\left(t \cdot \mathrm{tr}(A) - \frac{t^{2}}{2} \mathrm{tr}(A^{2}) + O(t^{3})\right) \\
&= 1 + t \mathrm{tr}(A) + \frac{t^{2}}{2}(\mathrm{tr}(A)^{2} - \mathrm{tr}(A^{2})) + O(t^{3}).
\end{aligned}
\]
◻
We thus obtain the first variation formula:
\[
\begin{aligned}
&\frac{d}{dt}|_{t=0} |\Sigma_t\cap U| = \frac{d}{dt}|_{t=0} \int_{\Sigma} \sqrt{\det(\left< e_i,e_j \right>)} \, d\Sigma \\
={}& \frac{1}{2} \int_{\Sigma} \mathrm{tr}(\left< e_i',e_j \right> + \left< e_i,e_j' \right> )_{ij} \, d\Sigma = n \int_{\Sigma} H \, d\Sigma\\
={}& \int_{\Sigma} \left< \nabla_{e_i}V,e_i \right> d\Sigma=\int_{ \Sigma} \mathrm{div}^\Sigma V d\Sigma
\end{aligned}
\]
For the second derivative of the area element, set \(M(t)=(\left< e_i(t),e_j(t) \right>)_{ij}\). Then
\[
\begin{aligned}
&\frac{d^2}{dt^2}|_{t=0} \sqrt{\mathrm{det}(M(t))}=\frac{1}{2}\frac{(\mathrm{det}\,M(t))''}{\sqrt{\mathrm{det}\,M(t)}}-\frac{1}{4}\frac{(\mathrm{det}\,M(t))'^2}{(\mathrm{det}\,M(t))^{3/2}}\\
={}&\frac{1}{2}(\mathrm{tr}\,M''+(\mathrm{tr}\,M')^{2}-\mathrm{tr}\,(M'^{2}))-\frac{1}{4}(\mathrm{tr}\,M')^{2}\\
={}&\frac{1}{2}\mathrm{tr}\,M''+\frac{1}{4}(\mathrm{tr}\,M')^{2}-\frac{1}{2}\mathrm{tr}\,(M'^{2}).
\end{aligned}
\]
We compute each term:
\[
\begin{aligned}
\frac{1}{2}\mathrm{tr}\,M''={}&\sum_{i=1}^n \left< \nabla_{V}\nabla_{V}e_i,e_i \right> + \sum_{i=1}^n \left< \nabla_{V}e_i,\nabla_{V}e_i \right>\\
={}&\sum_{i=1}^n-\left< R(V,e_i)V,e_i \right> + \left< \nabla_{e_i}\nabla_{V}V,e_i \right> + \sum_{i=1}^{n}\left< \nabla_{e_i}V,\nabla_{e_i}V \right> \\
={}&\sum_{i=1}^n-\left< R(V,e_i)V,e_i \right> + \mathrm{div}^\Sigma \nabla_{V}V + \sum_{i=1}^{n}|\nabla_{e_i} V|^{2},\\
\frac{1}{4}(\mathrm{tr}\,M')^{2}={}&\frac{1}{4}\left(\sum_{i=1}^n \left< \nabla_{V}e_i,e_i \right> + \left< e_i,\nabla_{V}e_i \right> \right)^{2}=\left(\sum_{i=1}^n \left< \nabla_{e_i}V,e_i \right> \right)^{2}=(\mathrm{div}^\Sigma V)^{2},\\
\frac{1}{2}\mathrm{tr}\,(M'^{2})={}&\frac{1}{2}\sum_{i,j=1}^n \left( \left< \nabla_{e_i}V, e_j\right> + \left< \nabla_{e_j}V, e_i \right> \right) ^{2} = \sum_{i,j=1}^n \left< \nabla_{e_i}V, e_j\right> ^{2} + \sum_{i,j=1}^n \left< \nabla_{e_i}V, e_j\right> \left< \nabla_{e_j}V, e_i \right>\\
={}&\sum_{i}^n |(\nabla_{e_i}V)^\top|^{2} + \sum_{i,j=1}^n \left< \nabla_{e_i}V, e_j\right> \left< \nabla_{e_j}V, e_i \right>.
\end{aligned}
\]
So
\[
\begin{aligned}
\frac{d^2}{dt^2}|_{t=0} \sqrt{\mathrm{det}(M(t))} = \sum_{i=1}^n-\left< R(V,e_i)V,e_i \right> + \mathrm{div}^\Sigma \nabla_{V}V + (\mathrm{div}^\Sigma V)^{2}\\
+ \sum_{i=1}^{n}|(\nabla_{e_i}V)^\bot|^{2} - \sum_{i,j=1}^n \left< \nabla_{e_i}V, e_j\right> \left< \nabla_{e_j}V, e_i \right>.
\end{aligned}
\]
Then, we have the second variation formula:
\[
\begin{aligned}
&\frac{d^2}{dt^2}|_{t=0} |\Sigma_t\cap U| = \int_\Sigma \frac{d^2}{dt^2}|_{t=0} \sqrt{\mathrm{det}(M(t))} \, d\Sigma \\
={}&\int_\Sigma \sum_{i=1}^n-\left< R(V,e_i)V,e_i \right> + \mathrm{div}^\Sigma \nabla_{V}V + (\mathrm{div}^\Sigma V)^{2} \\
& + \sum_{i=1}^{n}|(\nabla_{e_i}V)^\bot|^{2} - \sum_{i,j=1}^n \left< \nabla_{e_i}V, e_j\right> \left< \nabla_{e_j}V, e_i \right> \, d\Sigma.
\end{aligned}
\]
If \(V\) has compact support in \(U\), then by the divergence theorem
\[
\begin{aligned}
&\int_\Sigma \mathrm{div}^\Sigma V \, d\Sigma = \int_{ \Sigma}\mathrm{div}^\Sigma (V^\bot+V^\top) d\Sigma\\
={}&\int_{ \Sigma}\mathrm{div}^\Sigma V^\bot d\Sigma = \int_{ \Sigma} \left< \vec{H},V \right> d\Sigma.
\end{aligned}
\]
Definition 1.3.3
A submanifold \(\Sigma\) is called minimal if it is a critical point of the volume functional, i.e., \(\delta_V(U) = 0\) for all variations \(V\) with compact support in \(U\).
By the first variation formula, \(\Sigma\) is minimal if and only if \(\vec{H} = 0\).
Suppose \(\Sigma\) is minimal, and the variation vector field \(V\) is normal, i.e., \(V = V^\perp\), with compact support in \(U\). Then, we know \(\mathrm{div}^\Sigma V = 0\), and \(\int_{ \Sigma} \mathrm{div}^\Sigma \nabla_{V}V \, d\Sigma = 0\). So the second variation formula simplifies to
\[
\frac{d^{2}}{dt^{2}}\bigg|_{t=0} |\Sigma_t\cap U| = \int_\Sigma \sum_{i=1}^n-\left< R(V,e_i)V,e_i \right> + \sum_{i=1}^{n}|(\nabla_{e_i}V)^\bot|^{2} - \sum_{i,j=1}^n \left< \nabla_{e_i}V, e_j\right> \left< \nabla_{e_j}V, e_i \right> \, d\Sigma.
\]
Furthermore, if \(\Sigma\) is a two-sided hypersurface, then we can write \(V = f\nu\) for some smooth function \(f\) with compact support in \(U\). In this case, the second variation formula becomes
\[
\frac{d^{2}}{dt^{2}}\bigg|_{t=0} |\Sigma_t\cap U| = \int_\Sigma |\nabla f|^{2} - (|A|^{2} + \mathrm{Ric}(\nu,\nu))f^{2} \, d\Sigma,
\]
This formula is useful in the study of minimal hypersurfaces. For instance, it directly yields the second variation formula for minimal hypersurfaces with free boundary.
Proposition 1.3.4
Suppose \((\Sigma,\partial\Sigma)\hookrightarrow (M,\partial M)\) is a minimal hypersurface with free boundary, i.e., \(\Sigma\) meets \(\partial M\) orthogonally along \(\partial \Sigma\). Then the second variation formula for normal variations \(V = f\nu\) with compact support in \(U\) is
\[
\frac{d^{2}}{dt^{2}}\bigg|_{t=0} |\Sigma_t\cap U| = \int_\Sigma |\nabla f|^{2} - (|A|^{2} + \mathrm{Ric}(\nu,\nu))f^{2} \, d\Sigma - \int_{\partial \Sigma} A_{\partial M}(\nu,\nu)f^{2} \, d\sigma,
\]
Proof. It remains only to handle the term \(\int_{ \Sigma} \mathrm{div}^\Sigma \nabla_{V}V \, d\Sigma\) in the second variation formula. By the divergence theorem, we have
\[
\int_{ \Sigma} \mathrm{div}^\Sigma \nabla_{V}V \, d\Sigma = \int_{\partial \Sigma} \left< \nabla_{V}V, \eta \right> \, d\sigma=\int_{\partial \Sigma} A_{\partial M}(\nu,\nu)f^{2} \, d\sigma,
\]
◻
Definition 1.3.5
A minimal hypersurface \(\Sigma\) is called stable if the second variation of the volume functional is nonnegative for all variations with compact support, i.e., \(\frac{d^{2}}{dt^{2}}\bigg|_{t=0} |\Sigma_t\cap U| \geq 0\) whenever \(\mathrm{spt}\,V \subset U\).
If \(\Sigma\) is a two-sided stable minimal hypersurface, this is equivalent to the following stability inequality for all smooth functions \(f\) with compact support in \(U\):
\[
\int_\Sigma |\nabla f|^{2} - (|A|^{2} + \mathrm{Ric}(\nu,\nu))f^{2} \, d\Sigma \geq 0.
\]
Corollary 1.3.6
The variation of mean curvature \(H\) is given by
\[
\frac{d}{dt}H=-\Delta f - (|A|^{2} + \mathrm{Ric}(\nu,\nu))f,
\]
where \(V=f\nu\) is a normal variation with compact support in \(U\).
Proof. We consider only the normal variation \(V = f\nu\). The first variation gives
\[
\int_{ \Sigma} H f d\Sigma.
\]
Hence the second variation can be expressed as
\[
\int_{ \Sigma} H \frac{d}{dt}f + \frac{d}{dt}H f + (Hf)^{2} d\Sigma.
\]
Comparing this expression with the second variation formula, we conclude that
\[
\begin{aligned}
\int_{ \Sigma} \frac{d}{dt}H f={}&\int_{ \Sigma} -\sum_{i=1}^{n}\left< R(V,e_i)V,e_i \right> + \sum_{i=1}^{n}|(\nabla_{e_i}V)^\bot|^{2} - \sum_{i,j=1}^n \left< \nabla_{e_i}V, e_j\right> \left< \nabla_{e_j}V, e_i \right>\\
={}&\int_{ \Sigma} |\nabla f|^{2}- (|A|^{2} + \mathrm{Ric}(\nu,\nu))f^{2} d\Sigma\\
={}&\int_{ \Sigma} -f\left( \Delta f + (|A|^{2} + \mathrm{Ric}(\nu,\nu))f \right) d\Sigma
\end{aligned}
\]
Since this holds for all \(f\) with compact support in \(U\), we have
\[
\frac{d}{dt}H =-\Delta f - (|A|^{2} + \mathrm{Ric}(\nu,\nu))f.
\]
◻
Define the Jacobi operator \(L\) by
\[
L = -\Delta - (|A|^{2} + \mathrm{Ric}(\nu,\nu)).
\]
Then the stability condition can be written as
\[
\int_\Sigma f L f \, d\Sigma \geq 0 \quad \text{for all } f \text{ with compact support in } U.
\]
Definition 1.3.7
The Morse index of a minimal hypersurface \(\Sigma\) is the dimension of the space of smooth functions \(f\) with compact support in \(U\) such that
\[
\int_\Sigma f L f \, d\Sigma \lt{} 0,
\]
i.e., the maximum dimension of a subspace on which the quadratic form is negative definite.
Equivalently, the Morse index counts the number of negative eigenvalues (with multiplicity) of the Jacobi operator \(L\) under appropriate boundary conditions. A minimal hypersurface is stable if and only if its Morse index is zero.
Theorem 1.3.8
Let \(\Sigma^n\) be a complete minimal hypersurface in a Riemannian manifold \((M^{n+1}, g)\). If \(\Sigma\) has finite Morse index, then \(\Sigma\) is stable outside a compact set, i.e., there exists a compact set \(K \subset \Sigma\) such that
\[
\int_\Sigma |\nabla f|^{2} - (|A|^{2} + \mathrm{Ric}(\nu,\nu))f^{2} \, d\Sigma \geq 0
\]
for all \(f \in C_c^\infty(\Sigma \setminus K)\).
Proof. Let \(\text{ind}(\Sigma) = m \lt{} \infty\) be the Morse index of \(\Sigma\). By definition, there exists a finite-dimensional subspace \(V \subset C_c^\infty(\Sigma)\) of dimension \(m\) such that the quadratic form
\[
Q(f) = \int_\Sigma |\nabla f|^{2} - (|A|^{2} + \mathrm{Ric}(\nu,\nu))f^{2} \, d\Sigma
\]
is negative definite on \(V\), and \(Q(f) \geq 0\) for all \(f \perp V\) (in the \(L^2\) inner product sense).
Since each function \(f_i\) in a basis of \(V\) has compact support, there exists a compact set \(K \subset \Sigma\) such that \(\text{spt}(f_i) \subset K\) for all \(i = 1, \ldots, m\).
Now, for any \(f \in C_c^\infty(\Sigma \setminus K)\), we have \(\text{spt}(f) \cap K = \emptyset\). This means \(f\) is orthogonal to every basis function \(f_i\) of \(V\) (since their supports are disjoint). Therefore, \(f \perp V\), and by the definition of finite Morse index, we have
\[
Q(f) = \int_\Sigma |\nabla f|^{2} - (|A|^{2} + \mathrm{Ric}(\nu,\nu))f^{2} \, d\Sigma \geq 0.
\]
This proves that \(\Sigma\) is stable outside the compact set \(K\). ◻
