Minimal Submanifolds and First Variation

Throughout, manifolds and maps are smooth unless stated otherwise. On a Riemannian manifold, \(\nabla\) denotes the Levi–Civita connection. The needed local constructions start with coordinate charts.

A surface is stationary for area when every compactly supported deformation fixing its boundary has zero first variation. For a graph, this condition becomes a nonlinear differential equation; for a general submanifold, it becomes vanishing of the mean curvature vector.

Classical examples are the plane, the catenoid (Euler, 1741), and the helicoid (Meusnier, 1776). We derive both descriptions, beginning with graphs. A closed differential form then proves a stronger minimizing property for minimal graphs under the comparison hypotheses below. The final applications use harmonic coordinate functions and radial variations to obtain convex-hull and monotonicity results.

1. Minimal Surfaces

To start, let \(u : \Omega \subset \mathbb R^2 \to \mathbb R\) be a \(\mathscr C^2\) function. Define a surface in \(\mathbb R^3\), called the graph of \(u\), as

\[ \operatorname{Graph} (u) \triangleq \left\{ (x,y,z) \in \mathbb R^3 : z=u(x,y) \text{ for } (x,y) \in \Omega \right\} ~.\tag{1}\]

Define the area of a surface in the usual sense, by

\[ \operatorname{Area}(\operatorname{Graph}(u)) \triangleq \int_\Omega \left| \begin{pmatrix} 1 \\ 0 \\ u_x \end{pmatrix} \times \begin{pmatrix} 0 \\ 1 \\ u_y \end{pmatrix} \right| = \int_\Omega \sqrt{1 + |\nabla u|^2} ~, \tag{2}\]

and the unit normal vector is given by

\[ N = \frac{(-u_x,-u_y,1)}{\sqrt{1+|\nabla u|^2}} ~.\tag{3}\]

To characterize stationarity, take \(\eta\in C_c^2(\Omega)\), or on a compact domain impose \(\eta=0\) on its boundary. Then \(u+t\eta\) has the same boundary data. The area variation is finite on the support of \(\eta\), even if the entire graph has infinite area. Differentiation under the integral is justified by compact support and continuity of the derivatives.

\[ \operatorname{Area}(\operatorname{Graph}(u + t\eta)) = \int_\Omega \sqrt{1 + |\nabla u + t \nabla \eta|^2} ~.\tag{4}\]

We can then proceed to characterize the critical point of the area functional: direct calculations reveal

\[\begin{aligned} \dod{}{t}\Big|_{t=0} \operatorname{Area}(\operatorname{Graph}(u + t\eta)) = \int_\Omega \frac{\langle \nabla \eta, \nabla u \rangle}{\sqrt{1 + |\nabla u|^2}} = -\int_\Omega \eta \operatorname{div} \left( \frac{\nabla u}{\sqrt{1 + |\nabla u|^2}} \right) ~. \end{aligned}\tag{5}\]

If this first variation vanishes for every compactly supported test function \(\eta\), integration by parts and the fundamental lemma give the minimal surface equation below. Conversely, that equation makes every such first variation zero. Thus the equation characterizes stationarity, and the calibration argument supplies the stronger minimizing conclusion.

\[ \operatorname{div} \left( \frac{\nabla u}{\sqrt{1 + |\nabla u|^2}} \right) = 0\tag{6}\]

Thus \(u\) is stationary for compactly supported graph variations precisely when it satisfies (6), the minimal surface equation in divergence form. The test functions \(\eta\) are compactly supported in \(\Omega\), or vanish on the fixed boundary of the compact graph portion under consideration.

Stationarity of a minimal graph also yields a comparison with oriented surfaces in the same cylinder. Extend its upward unit normal \(N=(-u_x,-u_y,1)/W\), \(W=\sqrt{1+|\nabla u|^2}\), independently of the vertical coordinate. Define the two-form \(\omega(X,Y)=\det(X,Y,N)=\langle X\times Y,N\rangle\). Its coordinate components are

\[\omega(\partial_x,\partial_y)=W^{-1},\quad \omega(\partial_y,\partial_z)=-u_x/W,\quad \omega(\partial_x,\partial_z)=u_y/W,\]

\[\omega=\frac{dx\wedge dy-u_x\,dy\wedge dz+u_y\,dx\wedge dz}{W}.\]

Since \(\omega=\iota_N(dx\wedge dy\wedge dz)\),

\[d\omega=(\operatorname{div}_{\mathbb R^3}N)\,dx\wedge dy\wedge dz =-\operatorname{div}_{\mathbb R^2}\left(\frac{\nabla u}{W}\right)dx\wedge dy\wedge dz=0.\]

For an orthonormal pair \(X,Y\), \(|\omega(X,Y)|\leq1\) by Cauchy–Schwarz. Equality in absolute value means that their plane is \(N^\perp\); the value is \(+1\) for the orientation whose normal is \(N\). In particular, \(\omega\) restricts to the area form on the upward-oriented graph. A closed form with this norm bound is called a calibration.

Theorem 1 (Minimal graph calibration). Let \(u\in C^2(\Omega)\) solve the minimal surface equation. Let \(\Gamma\) be a compact oriented portion of its graph, and let \(\Sigma\) be a compact oriented comparison surface in \(\Omega\times\mathbb R\), with the same oriented boundary. Assume \(\Sigma-\Gamma\) bounds an oriented three-chain in the cylinder, so the surfaces are homologous there relative to their common boundary. Then

\[\operatorname{Area}(\Gamma)\leq\operatorname{Area}(\Sigma).\]

The chain may be piecewise smooth. This formulation makes the finite-area and homology conditions used by Stokes explicit; it applies in particular to comparisons obtained by a compact deformation fixing the boundary.

Proof. If \(\Sigma-\Gamma=\partial C\), the closedness calculation gives

\[\int_\Sigma\omega-\int_\Gamma\omega=\int_Cd\omega=0.\]

The graph orientation makes \(\int_\Gamma\omega=\operatorname{Area}(\Gamma)\). On the comparison surface the pointwise bound \(\omega(e_1,e_2)\leq1\), integrated against its area element, gives

\[\operatorname{Area}(\Gamma)=\int_\Sigma\omega\leq\operatorname{Area}(\Sigma).\]

Equality requires agreement with the calibrated oriented plane almost everywhere. This proof compares all the specified homologous competitors, rather than just infinitesimal variations. ◻

2. Minimal Submanifolds

Let \(\Sigma^k\subset(M^n,g)\) be a smooth immersed submanifold, with induced metric. We use the ambient Levi–Civita connection, not an arbitrary affine connection. For tangent vector fields \(X,Y\), write

\[\nabla^\Sigma_XY=(\nabla^M_XY)^\top,\qquad B(X,Y)=(\nabla^M_XY)^\perp.\]

Here \(B\) is the vector-valued second fundamental form, also denoted \(\operatorname{II}\) in the graph calculations. Torsion-freeness gives \(B(X,Y)=B(Y,X)\), because \([X,Y]\) is tangent. For a local orthonormal normal frame \(N_\alpha\), metric compatibility gives

\[B(X,Y)=\sum_\alpha\langle\nabla^M_XY,N_\alpha\rangle N_\alpha =-\sum_\alpha\langle Y,\nabla^M_XN_\alpha\rangle N_\alpha.\]

With a tangent orthonormal frame \(E_1,\ldots,E_k\), set

\[|B|^2=\sum_{i,j=1}^k|B(E_i,E_j)|^2,\qquad H=\sum_{i=1}^k B(E_i,E_i).\]

Thus our mean curvature vector is the trace, without a factor \(1/k\). The Gauss equation reads

\[(K_\Sigma(X,Y)-K_M(X,Y))|X\wedge Y|^2 =\langle B(X,X),B(Y,Y)\rangle-|B(X,Y)|^2.\]

Remark 2. On a hypersurface with a chosen local unit normal \(N\), our shape operator is \(S_NX=\nabla^M_XN\). Its eigenvalues are \(\kappa_i\). Since \(\langle B(X,Y),N\rangle=-\langle S_NX,Y\rangle\),

\[\langle H,N\rangle=-\sum_{i=1}^{n-1}\kappa_i.\]

This agrees with the earlier convention \(\operatorname{II}_N(X,Y)=\langle S_NX,Y\rangle\): that scalar form is the negative of the normal component of the vector-valued \(B\). A global unit normal is unnecessary in higher codimension or for the first variation formula.

For any vector field \(V\) along \(\Sigma\), including one with a normal component, define its tangential divergence by

\[\operatorname{div}_\Sigma V=\sum_{i=1}^k\langle\nabla^M_{E_i}V,E_i\rangle,\qquad \Delta_\Sigma f=\operatorname{div}_\Sigma(\operatorname{grad}_\Sigma f).\tag{7}\]

The first expression equals intrinsic divergence when \(V\) is tangent. For a normal field,

\[\operatorname{div}_\Sigma V^\perp =-\sum_i\langle V^\perp,\nabla^M_{E_i}E_i\rangle=-\langle V,H\rangle.\]

Consequently

\[\operatorname{div}_\Sigma V=\operatorname{div}_\Sigma V^\top-\langle V,H\rangle.\]

The minus sign follows from differentiating \(\langle V^\perp,E_i\rangle=0\).

Let \(\Gamma:(-\epsilon,\epsilon)\times\Sigma\to M\) be a variation through immersions, with \(\Gamma(0,x)=x\), compact support in the spatial variable, and fixed boundary if a boundary is present. Put \(V=\partial_t\Gamma|_{t=0}\). In local coordinates \(x_1,\ldots,x_k\), its induced metric and relative volume density are

\[g_{ij}(t)=\langle\partial_i\Gamma,\partial_j\Gamma\rangle,\qquad \nu(t)=\frac{\sqrt{\det g(t)}}{\sqrt{\det g(0)}}.\]

Both determinants acquire the same squared Jacobian under a coordinate change, so the ratio is intrinsic. The varied volume is \(\int_\Sigma\nu(t)\,d\mu_\Sigma\); for noncompact \(\Sigma\), differentiate only its finite change on the compact support.

Jacobi's determinant identity and compatibility of the connection give

\[\nu'(0)=\frac12g^{ij}(0)\,\partial_tg_{ij}(0),\qquad \partial_tg_{ij}(0)=\langle\nabla^M_{\partial_i}V,\partial_j\rangle+ \langle\partial_i,\nabla^M_{\partial_j}V\rangle.\]

The interchange \(D_t\partial_i\Gamma=D_i\partial_t\Gamma\) uses torsion-freeness of the pullback connection. Choose coordinates orthonormal at the point being evaluated. Then

\[\nu'(0)=\sum_{i=1}^k\langle\nabla^M_{E_i}V,E_i\rangle=\operatorname{div}_\Sigma V.\tag{8}\]

\[\nu'(0)=\operatorname{div}_\Sigma V^\top-\langle V,H\rangle.\tag{9}\]

This is a pointwise density identity. Integrating, and applying the intrinsic divergence theorem to the tangential part, gives the first variation formula

\[\left.\frac d{dt}\right|_{0}\operatorname{Vol}(\Gamma_t(\Sigma)) =-\int_\Sigma\langle V,H\rangle\,d\mu_\Sigma +\int_{\partial\Sigma}\langle V,\eta\rangle\,d\sigma,\tag{10}\]

where \(\eta\) is the outward unit conormal in \(\Sigma\). The boundary term vanishes for a fixed-boundary variation.

Thus \(H=0\) makes every such first variation zero. Conversely, every smooth compactly supported normal field in the interior can be realized by \(\Gamma(t,x)=\exp_x(tV(x))\) for sufficiently small \(t\). If \(H\) were nonzero at an interior point, choosing \(V=\chi H\) with a nonnegative bump \(\chi\) there would give a strictly negative first variation. Hence stationarity is equivalent to \(H\equiv0\), and continuity includes any boundary points. Such a submanifold is called minimal.

Equivalently, for all compactly supported ambient fields along \(\Sigma\) that vanish at its boundary,

\[\int_\Sigma\operatorname{div}_\Sigma V\,d\mu_\Sigma=0.\tag{11}\]

The fields must allow normal components: testing only tangent fields would merely repeat the divergence theorem on every submanifold. This formulation can later be used to define stationarity for less regular geometric objects.

3. Consequences of the First Variation Formula

We next derive consequences of (11). These will culminate in the theorem of Bernstein, treated in the next section. We look predominantly at minimal submanifolds \(\Sigma^k \subset \mathbb R^n\), where \(k < n\). We drop the dependence of \(\Sigma\) on dimension \(k\); unless otherwise stated, the \(\dim \Sigma = k\).

Proposition 3. \(\Sigma \subset \mathbb R^n\) is minimal if and only if the restrictions of the coordinate functions of \(\mathbb R^n\) to \(\Sigma\) are harmonic functions.

In fact the calculation proves the pointwise identity \(\Delta_\Sigma x_i=\langle H,e_i\rangle\). Equality against every compactly supported test function implies equality pointwise, since both sides are continuous. The ambient coordinate vectors span the whole normal as well as tangent space, so vanishing of all coordinate Laplacians is equivalent to \(H=0\).

Proof. Let \(e_i = \nabla_{\mathbb R^n} x_i = \pd{}{{x_i}}\) denote the \(i\)th coordinate vector field on \(\mathbb R^n\), and \(\eta\) denote a smooth, compactly supported function on \(\Sigma\) that vanishes on \(\partial \Sigma\). From the fact that \(e_i\) is constant vector field, it follows

\[\operatorname{div}_\Sigma (\eta e_i) = \langle \nabla_\Sigma \eta, e_i \rangle = \langle \nabla_\Sigma \eta, \nabla_\Sigma x_i \rangle ~.\]

In other words,

\[\begin{aligned} \int_\Sigma \operatorname{div}_\Sigma (\eta e_i) &= - \int_\Sigma \langle \eta e_i, H \rangle = \int_\Sigma \langle \nabla_\Sigma \eta, \nabla_\Sigma x_i \rangle = - \int_\Sigma \eta \Delta_\Sigma x_i ~. \end{aligned}\]

The conclusion follows directly. ◻

Recall that if \(\Sigma \subset \mathbb R^n\) is a compact subset, then the smallest convex set containing \(\Sigma\), the convex hull of \(\Sigma\), denoted \(\operatorname{Conv}(\Sigma)\), is the intersection of all half-spaces containing \(\Sigma\). The maximum principle for harmonic coordinate functions gives the convex-hull property.

Proposition 4. If \(\Sigma \subset \mathbb R^n\) is a compact minimal submanifold, then \(\Sigma \subset \operatorname{Conv}(\partial \Sigma)\).

Assume positive dimension and compactness as a manifold with boundary. A compact connected component without boundary cannot be minimally immersed in Euclidean space: every coordinate function would be harmonic and therefore constant on that component, contradicting immersion. Thus the maximum principle applies with a nonempty boundary on each component. Applying the stated half-space argument for every \(e,a\) containing the boundary gives the intersection of all those half-spaces, which is precisely its closed convex hull.

Proof. Recall the half-space of \(\mathbb R^n\) can be written as

\[H \triangleq \{x \in \mathbb R^n : \langle x , e \rangle \leq a \} ~,\]

where \(a \in \mathbb R\) and \(e \in \mathbb S^{n-1}\). Then, the function \(\varphi(x) = \langle e, x \rangle\) is harmonic by Proposition 3, hence—by the maximum principle—\(\varphi(x)\) attains its maximum on \(\partial \Sigma\). ◻

Remark 5. A supporting hyperplane for a compact minimal submanifold meets its boundary. Interior contact is possible only if the corresponding linear function is constant on that connected component, in which case the entire component lies in the hyperplane. Thus “first contact must be on the boundary” does not exclude simultaneous interior contact in the planar case.

Proposition 6 (The Monotonicity Formula). Suppose \(\Sigma \subset \mathbb R^n\) is a minimal submanifold, and \(x_0 \in \mathbb R^n\). Then, for all \(0 < s < t\),

\[\begin{aligned} \frac{\operatorname{Vol}(B_t(x_0)\cap \Sigma)}{t^k} - \frac{\operatorname{Vol}(B_s(x_0)\cap \Sigma)}{s^k} = \int_{(B_t(x_0)\setminus B_s(x_0)) \cap \Sigma} \frac{|(x-x_0)^\perp|^2}{|x-x_0|^{k+2}} ~. \end{aligned}\tag{12}\]

Here \(\Sigma^k\) is a smooth properly immersed minimal submanifold in a neighborhood of \(\overline B_R(x_0)\), with no boundary there, and \(0<s<t<R\). Volume counts immersion multiplicity; in the embedded case it is the usual induced volume. Properness ensures compact integration domains. Boundary terms are required if the balls meet \(\partial\Sigma\), so the displayed formula is not asserted across such a boundary.

Proof. Put \(z=x-x_0\), \(r=|z|\), and \(A(\rho)=\operatorname{Vol}(\Sigma\cap B_\rho(x_0))\). Since \(D_Ez=E\) for every ambient vector,

\[\nabla_\Sigma r^2=2z^\top,\qquad \operatorname{div}_\Sigma z=k,\qquad \Delta_\Sigma r^2=2\operatorname{div}_\Sigma z^\top=2k.\]

The last equality uses \(\operatorname{div}_\Sigma z^\perp=-\langle z,H\rangle=0\). Away from \(z=0\),

\[\nabla_\Sigma r=z^\top/r,\qquad |\nabla_\Sigma r|=|z^\top|/r.\]

For a regular radius \(\rho\), the outward conormal of \(\Sigma\cap B_\rho\) is \(\nabla_\Sigma r/|\nabla_\Sigma r|\). The divergence theorem gives

\[kA(\rho)=\rho\int_{\Sigma\cap\partial B_\rho}|\nabla_\Sigma r|\,d\sigma =\int_{\Sigma\cap\partial B_\rho}|z^\top|\,d\sigma.\]

Coarea gives, for almost every radius,

\[A'(\rho)=\int_{\Sigma\cap\partial B_\rho}\frac1{|\nabla_\Sigma r|}\,d\sigma =\int_{\Sigma\cap\partial B_\rho}\frac{\rho}{|z^\top|}\,d\sigma.\]

Combining these two identities, with all powers of \(\rho\) retained, yields

\[\begin{aligned} \frac d{d\rho}\bigl(\rho^{-k}A(\rho)\bigr) &=\rho^{-k-1}\int_{\Sigma\cap\partial B_\rho} \left(\frac{\rho^2}{|z^\top|}-|z^\top|\right)d\sigma\\ &=\rho^{-k-1}\int_{\Sigma\cap\partial B_\rho} \frac{|z^\perp|^2}{|z^\top|}\,d\sigma. \end{aligned}\tag{13}\]

Integrate from \(s\) to \(t\). Coarea cancels the factor \(|\nabla_\Sigma r|=|z^\top|/r\), giving

\[t^{-k}A(t)-s^{-k}A(s) =\int_{\Sigma\cap(B_t\setminus B_s)} \frac{|z^\perp|^2}{r^{k+2}}\,d\mu_\Sigma.\]

To justify the almost-everywhere operations, the critical set of \(r\) away from zero has \(k\)-dimensional measure zero. Indeed, at almost every density point of the zero set of the smooth vector field \(\nabla_\Sigma r^2\), its derivative vanishes: difference quotients along points of that zero set determine every direction. Its trace would then be zero, contradicting \(\Delta_\Sigma r^2=2k>0\). Coarea on the complement therefore shows that \(A\) is absolutely continuous on compact radius intervals away from zero. Sard's theorem supplies regular radii of full measure. Each level set has zero \(k\)-volume by the same argument, so continuity extends the integrated identity to every \(0<s<t<R\). The right side is nonnegative, proving monotonicity as well as the exact formula. ◻

Proposition 7 (Weighted monotonicity identity). Under the geometric hypotheses of Proposition 6, let \(f\in C^2(\Sigma)\), put \(z=x-x_0,\ r=|z|\), and let \(0<s<t<R\). Then

\[\begin{aligned} t^{-k}\int_{\Sigma\cap B_t(x_0)}f\,d\mu-s^{-k}\int_{\Sigma\cap B_s(x_0)}f\,d\mu &=\int_{\Sigma\cap(B_t(x_0)\setminus B_s(x_0))} f\,\frac{|z^\perp|^2}{r^{k+2}}\,d\mu\\ &\quad+\frac12\int_s^t\rho^{-k-1} \int_{\Sigma\cap B_\rho(x_0)}(\rho^2-r^2)\Delta_\Sigma f\,d\mu\,d\rho. \end{aligned}\tag{14}\]

In particular, the normalized integrals are nondecreasing if \(f\geq0\) and \(\Delta_\Sigma f\geq0\).

Proof. For a regular radius \(\rho\), apply Green's identity to \(f\) and \(r^2\), using \(\Delta_\Sigma r^2=2k\), \(r^2=\rho^2\) on the boundary, and \(\int_{\partial D}\partial_\eta f=\int_D\Delta_\Sigma f\). This gives

\[2k\int_{\Sigma\cap B_\rho}f =2\rho\int_{\Sigma\cap\partial B_\rho}f|\nabla_\Sigma r| -\int_{\Sigma\cap B_\rho}(\rho^2-r^2)\Delta_\Sigma f.\]

Let \(G(\rho)=\rho^{-k}\int_{\Sigma\cap B_\rho}f\). Weighted coarea and this identity give, almost everywhere,

\[G'(\rho)=\rho^{-k-1}\int_{\Sigma\cap\partial B_\rho} f\,\frac{|z^\perp|^2}{|z^\top|} +\frac12\rho^{-k-1}\int_{\Sigma\cap B_\rho}(\rho^2-r^2)\Delta_\Sigma f.\]

Integrating the first term by coarea gives precisely the annular integral in the statement. The second term already has the required form. Compactness of the spatial domains and \(C^2\) regularity justify the weighted integrals and their absolute continuity, using the same zero-measure critical set as in Proposition 6. Notice that the annuli are level sets of the radial distance \(r\), not of the arbitrary test function \(f\). ◻

Corollary 8 (Mean value estimate). Let \(\Sigma^k\) be smoothly embedded and minimal near \(\overline B_s(x_0)\), with compact intersection there and no boundary in a neighborhood of that closed ball, and let \(x_0\in\Sigma\). If \(f\in C^2\) is nonnegative and

\[\Delta_\Sigma f\geq-\lambda s^{-2}f,\qquad \lambda\geq0,\]

then, writing \(\omega_k=\operatorname{Vol}(B_1\subset\mathbb R^k)\),

\[f(x_0)\leq\frac{e^{\lambda/4}}{\omega_k s^k} \int_{\Sigma\cap B_s(x_0)}f\,d\mu \leq\frac{e^{\lambda/2}}{\omega_k s^k}\int_{\Sigma\cap B_s(x_0)}f\,d\mu.\tag{15}\]

Thus the original stated constant \(e^{\lambda/2}\) is valid, and the calculation gives the stronger \(e^{\lambda/4}\).

Proof. Set \(G(t)=t^{-k}\int_{\Sigma\cap B_t(x_0)}f\). The boundary term in the derivative identity of Proposition 7 is nonnegative. Since \(0\leq t^2-r^2\leq t^2\) on \(B_t(x_0)\),

\[G'(t)\geq-\frac{\lambda}{2s^2}t^{-k-1} \int_{\Sigma\cap B_t(x_0)}(t^2-r^2)f \geq-\frac{\lambda t}{2s^2}G(t).\]

Hence \(e^{\lambda t^2/(4s^2)}G(t)\) is nondecreasing. This integrating-factor argument does not divide by \(G\), so it also applies when \(f\) vanishes. Smoothness of the embedded submanifold and continuity of \(f\) give

\[\lim_{t\downarrow0}G(t)=\omega_k f(x_0)\]

by expressing \(\Sigma\) locally as a graph over its tangent plane, whose derivative vanishes at \(x_0\). Compare this limit with the value at \(s\) to obtain the estimate. ◻

Carmo, Manfredo Perdigão do. 1992. Riemannian Geometry. Mathematics. Theory & Applications. Boston: Birkhäuser.

Reading: Carmo (1992), Riemannian geometry.