Characterizing classical minimal surfaces via the entropy differential

1 Introduction

Let \(\Sigma \subset \mathbb{R}^3\) be a smooth, oriented minimal surface. In this paper, we introduce a meromorphic quadratic differential \(P\) on \(\Sigma\), which we call the entropy differential. We use \(P\) to characterize several classical surfaces – including Enneper’s surface, the catenoid and the helicoid. In particular, subsets of Enneper’s surface are the only minimal surfaces on which \(P\) vanishes – a fact which we use to prove a novel curvature estimate for embedded minimal surfaces with small entropy differential and an associated compactness result.

The differential \(P\) arises naturally in a number of different contexts. Of particular interest is the realization of \(T=\operatorname{Re}P\), which we call the entropy form, as a conservation law for the diffeomorphism invariant functional \[\mathcal{E}[g]=\int_{\Sigma} K_g \log K_g \mu_g.\] This functional, which is a type of entropy for the curvature, has been previously considered by R. Hamilton in the context of the Ricci flow on surfaces  [14]. In particular, we show that if \(g\) is a minimal surface metric (i.e. the metric induced by a smooth minimal immersion) for which \(K_g\neq 0\), then the metric \(\hat{g}=(-K_g)^{3/4} g\) is a critical point of \(\mathcal{E}\) with respect to compactly supported conformal deformations. The crucial fact used here is the observation – due to Ricci [22] – that such minimal surface metrics satisfy the so-called Ricci condition: \[\Delta_g \log |K_g|=4K_g.\] The differential \(P\) also arises as a certain geometric Schwarzian derivative of the Gauss map – a point of view which has antecedents in [9, 12] – and which we will study more thoroughly in a forthcoming paper [3].

A key observation of the present paper is that, modulo rigid motions, a minimal surface is determined, up to a three-parameter family, by its Hopf differential \(Q\) and its entropy differential \(P\). This allows one to characterize several classical minimal surfaces in terms of simple relationships between the Hopf and entropy differentials:

The families \(\mathcal{C}\) and \(\mathcal{H}\) are, respectively, the deformed catenoids and deformed helicoids. These are one parameter families of surfaces containing, respectively, the catenoid and the helicoid – their geometry is discussed thoroughly in Section 4.

A consequence of Item [IntroEnneperCase] of Theorem 1.1 are a family of novel curvature estimates for embedded minimal surfaces. Namely, we introduce a certain family of scale invariant quantities which measure the size of the entropy form and use standard blow-up arguments to derive curvature bounds for embedded surfaces for which these quantities are small. Specifically, for a constant \(\alpha>0\) and smooth minimal surface \(\Sigma\) with entropy form \(T\), we define: \[||T||_{\Sigma, \alpha}:=2^{\frac{1}{2(1+\alpha)}}\int_{\Sigma} |T|^{\frac{1}{1+\alpha}}_g |K_g|^{\frac{\alpha}{\alpha+1}} \mu_g.\] We justify this family by noting that, on the one hand they are scale invariant and, on the other, the “endpoints” are very natural. Indeed, \[\lim_{\alpha\to \infty} ||T||_{\Sigma, \alpha}=\int_{\Sigma} |K_{g}| \mu_g,\] i.e., one endpoint is the total Gauss curvature, a well studied quantity in minimal surface theory. While, \[\lim_{\alpha\to 0} ||T||_{\Sigma, \alpha}=\sqrt{2} \int_{\Sigma} |T|_g \mu_g,\] i.e., the other endpoint is the \(L^1\) norm of the entropy form which is invariant under the standard action of \(\rm PSL(2,\mathbb{C})\) on the Gauss map of \(\Sigma\), see [3]. We will not deal directly with this quantity due to the fact that the presence of umbilic points tends to make it infinite.

As these quantities are scale invariant, standard blow-up arguments give the following curvature estimates:

In Section 5.2, we address the question of the best possible \(\epsilon\) and some partial results are obtained. In particular, we obtain the following identity:

This suggests that as \(\alpha\to 0\) \[\epsilon(\alpha)=\frac{2\pi}{\alpha} +o\left( \frac{1}{\alpha}\right).\] which would be sharp on the catenoid. Using standard techniques, we observe that our curvature estimate gives a corresponding compactness results which we record in Theorem 5.10. We conclude the paper with Section 6, wherein the entropy form is used to make a connection between minimal surfaces in \(\mathbb R^3\) and gradient Ricci soliton metrics on surfaces.

Acknowledgements

The authors would like to thank Rob Kusner and Daniel Fox for several stimulating discussions regarding the topics of this paper. The authors are also grateful to the anonymous referee for carefully reading the article and many useful suggestions. The first author was partially supported by the NSF grants DMS-0902721 and DMS-1307953 and by the EPSRC Programme Grant “Singularities of Geometric Partial Differential Equations” number EP/K00865X/1. The second author was partially supported by the Mathematical Sciences Research Institute in Berkeley and by the Forschungsinstitut für Mathematik at ETH Zürich.

2 A Geometric Entropy Functional for Surfaces

2.1 Definitions

We assume \(\mathbb{R}^3\) to be equipped with the standard Euclidean metric \(g_E\) and orientation. Let \(B_{R}(p)\) be the open Euclidean ball in \(\mathbb{R}^3\) with radius \(R>0\) and center \(p\). If \(p\) is omitted then the ball is assumed to be centered at the origin in \(\mathbb{R}^3\). Let \(M\) be an open orientable smooth \(2\)-manifold. For a smooth immersion \(\mathbf{x}: M \to \mathbb{R}^3\) let \(\Sigma=\mathbf{x}(M)\), we say \(\Sigma\) is properly embedded if \(\mathbf{x}\) is proper and injective. Moreover, let \(g=\mathbf{x}^*g_E\) be the first fundamental form. We write \(\tensor[^g]{\nabla}{}\) for the Levi-Civita connection, \(K_g\) for the Gauss curvature, and \(\mu_g\) for the area form of \(g\). The integrable almost complex structure on \(M\) induced by \(g\) and the orientation will be denoted by \(J\). Furthermore, for \(A \in \Gamma(S^2(T^*M))\), we define \((JA)(X,Y)=A(JX,Y)\) where \(X,Y \in \Gamma(TM)\) are smooth vector fields on \(M\). Here, as usual, \(S^2(T^*M)\) denotes the second symmetric power of the cotangent bundle of \(M\). In particular, the map \(A\mapsto A+i JA\) embeds the space of symmetric trace-free \(2\)-forms on \(M\) into the space of quadratic differentials on \(M\). Furthermore, we use the standard fact that \(A+iJA\) is holomorphic if and only if \(A\) is divergence-free.

Let \(\mathbf{n}\) denote the orientation compatible Gauss map of \(\mathbf{x}\) taking values in \(\partial B_1\), the unit-sphere in \(\mathbb{R}^3\) centered at \(0\). The second fundamental form of \(\mathbf{x}\) will be denoted by \(A\) and its trace with respect to \(g\), the mean curvature, by \(H\). A point \(p\in M\) at which the eigenvalues of \(A_p\) agree is called umbilic and we define \(\hat{M}\subset M\) to be the open submanifold of non-umbilic points.

The pair \((g,A)\) satisfies the Gauss equation \[|A|_g^2+2K_g =(\operatorname{tr}_g\! A)^2\] and the Codazzi equations \[\tensor[^g]{\nabla}{}_X A (Y,Z)=\tensor[^g]{\nabla}{}_Y A(X,Z)\] where \(X,Y,Z\in \Gamma(TN)\). Conversely, Bonnet’s theorem states that if a pair \((g,A)\) on a simply connected surface \(N\) satisfies the Gauss - and Codazzi equations, then there exists an immersion \(\mathbf{x}: N \to \mathbb{R}^3\) – unique up to composition with a rigid motion of \(\mathbb{R}^3\) – whose first and second fundamental form are \(g\) and \(A\). For this reason we refer to the triple \((M,g,A)\) as geometric data of \(\mathbf{x}\).

2.2 The Ricci condition

We suppose from now on that \(\mathbf{x}: M \to \mathbb{R}^3\) is minimal. That is \(H\equiv 0\). The Gauss equations imply that \(K_g\leq 0\) and that \(K_g\) is negative on \(\hat{M}\). It follows from the Codazzi equations that the second fundamental form is divergence free with respect to \(g\). This yields Simons’ identity \[\Delta_g A =-|A|_g^2 A\] where here \(\Delta_g\) is the rough Laplacian. From \[4 |A|_g^2 |\tensor[^g]{\nabla}{}A|^2_g=|\tensor[^g]{\nabla}{}|A|_g^2|^2,\] Simons’ identity, and the Gauss equations we obtain that on \(\hat{M}\) the following Ricci condition \[ \Delta_g \log |{K}_g| = 4 K_g\](2.1) holds. Abbreviate \[ u_g=-\frac{1}{4}\log|K_g|,\](2.2) then the Ricci condition becomes \[\Delta_gu_g=e^{-4u_g}.\]

Conversely, Ricci [22] showed that if \(g\) is a Riemannian metric of strictly negative Gauss curvature \(K_g\) on a simply connected \(2\)-manifold \(N\) satisfying the Ricci condition, then there exists a minimal immersion \(\mathbf{x}: N \to \mathbb{R}^3\) with \(\mathbf{x}^*g_E=g\). A proof of this fact using modern language may be found in [6].

2.3 The entropy functional

We will study a certain natural functional \(\mathcal{E}\) defined on the space \(\mathcal{M}^+(M)\) of smooth Riemannian metrics on \(M\) which have positive Gauss curvature. Define for \(g\in \mathcal{M}^+(M)\) \[\mathcal{E}[g]= \int_M K_g \log K_g \mu_g\] This functional has been applied to the study of Ricci flow on surfaces by Hamilton [14] and Chow [7] – in particular Hamilton observed that it is monotonically increasing along the Ricci flow on spheres with positive Gauss curvature (see Section 6 of the present paper for additional connections to Ricci solitons).

We compute the Euler-Lagrange equations associated to \(\mathcal{E}\). To do so, let \(f\) be a smooth symmetric \(2\)-form and write \(F=\operatorname{tr}_g f\) for its trace with respect to \(g\). Let \(g_t=g+tf\), then (cf. [8]) \[\begin{aligned} \left.\frac{\partial}{\partial t}\right|_{t=0} K_{g_t} &= - \frac{1}{2} \Delta_g F + \frac{1}{2} \operatorname{div}_g(\operatorname{div}_g\! f)- \frac{1}{2} K_g F\\ &= - \frac{1}{4} \Delta_g F + \frac{1}{2} \operatorname{div}_g\left(\operatorname{div}_g\! \mathring{f}\right)- \frac{1}{2} K_g F \end{aligned}\] where \(\mathring{f}\) is the trace-free part of \(f\). Hence, \[\left.\frac{\partial}{\partial t}\right|_{t=0} K_{g_t} \log K_{g_t} = \left(-\frac{1}{4} \Delta_g F + \frac{1}{2} \operatorname{div}_g\left(\operatorname{div}_g\! \mathring{f}\right)- \frac{1}{2} K_g F\right) (\log K_{g_t}+1)\] and so \[\delta_f\mathcal{E}[g]= \frac{1}{2}\int_{M}F K_g \log K_g+\left(\operatorname{div}_g\left(\operatorname{div}_g\! \mathring{f}\right)-\frac{1}{2} \Delta_g F - K_g F\right) (\log K_{g_t}+1)\mu_g.\] If \(f\) is compactly supported then using Green’s formula and the divergence theorem, i.e., integrating by parts twice, yields \[\delta_f\mathcal{E} [g]= -\frac{1}{4}\int_{M} F \left(\Delta_g \log K_g +2 K_g \right) -2\langle f, \tensor[^g]{\mathring{\nabla}}{}^2 \log K_g\rangle_g \mu_g,\] where \(\tensor[^g]{\mathring{\nabla}}{}^2\) denotes the trace-free Hessian and \(\langle a,b\rangle_g\) the natural bilinear pairing on elements \(a,b\in \Gamma(S^2(T^*M))\) obtained via \(g\).

We will say that \(g\in \mathcal{M}^+(M)\) is an \(\mathcal{E}\)-critical metric if \(\mathcal{E}\) is stationary at \(g\) with respect to compactly supported conformal deformations. Hence, \(g\) is an \(\mathcal{E}\)-critical metric if and only if the Gauss curvature \(K_g\) of \(g\) satisfies \[\Delta_g \log K_g = -2 K_g.\]

As \(\mathcal{E}\) is computed purely in terms of geometric quantities it is manifestly diffeomorphism invariant, that is if \(\phi: M \to M\) is a diffeomorphism we have \[\mathcal{E}[\phi^* g] =\mathcal{E}[g].\] By Noether’s principle this invariance leads to a conservation law for \(\mathcal{E}\)-critical metrics. Indeed, let \(X\) be a compactly supported vector field on \(M\) and \(\phi_t\) the flow of \(X\). We have that \[\phi_t^* g=g+ t L_X g +o(t).\] Recall, the Lie derivative, \(L_X g\) can be computed as \[(L_X g)(Y,Z)=g( \tensor[^g]{\nabla}{}_Y X,Z) + g(\tensor[^g]{\nabla}{}_Z X, Y).\] where \(Y,Z \in \Gamma(TM)\). By the diffeomorphism invariance we have at an \(\mathcal{E}\)-critical metric that \[\begin{aligned} 0 &= \delta_{L_X g} \mathcal{E} [g] \\ &= \frac{1}{2} \int_{M} \langle L_X g, \tensor[^g]{\mathring{\nabla}}{}^2 \log K_g\rangle_g \mu_g\\ &=\int_{M} \operatorname{div}_g\! \left( (\tensor[^g]{\mathring{\nabla}}{}^2 \log K_g )(X, \cdot)\right) - \langle X, \operatorname{div}_g\! \tensor[^g]{\mathring{\nabla}}{}^2 \log K_g \rangle_g \mu_g\\ &=-\int_{M} \langle X, \operatorname{div}_g\! \tensor[^g]{\mathring{\nabla}}{}^2 \log K_g \rangle_g \mu_g \end{aligned}\] where we used that \(X\) has compact support and the divergence theorem. As \(X\) is arbitrary, \[ \operatorname{div}_g\! \tensor[^g]{\mathring{\nabla}}{}^2 \log K_g =0.\](2.3) In other words, the quantity \[V_g=\tensor[^g]{\mathring{\nabla}}{}^2 \log K_g\] is a trace-free divergence free symmetric \(2\)-form, i.e. a conservation law for the \(\mathcal{E}\) functional.

2.4 The entropy form

Let \(g\) be a smooth Riemannian metric and \(\omega\) a smooth real-valued function on \(M\). We note the following standard formula for the trace-free Hessian and the Laplacian operating on \(u\in C^{\infty}(M)\) \[ \begin{aligned} \Delta_{e^{2\omega}g}u&=e^{-2\omega}\Delta_gu,\\ ^{e^{2\omega}g}\mathring{\nabla}^2&u=\tensor[^g]{\mathring{\nabla}}{}^2 u-\left(\mathrm{d}u \otimes \mathrm{d}\omega+\mathrm{d}\omega \otimes \mathrm{d}u-g(\tensor[^g]{\nabla}{}u,\tensor[^g]{\nabla}{}\omega)g\right). \end{aligned}\](2.4) Also, the Gauss-curvature transforms under conformal change as \[K_{e^{2\omega}g}=e^{-2\omega}\left(K_g-\Delta_g \omega \right).\] We let \(\mathcal{R}^{\pm}_{C}\) denote the space of smooth positively (or negatively) curved Riemannian metrics on \(M\) satisfying the generalized Ricci condition \[\Delta_g\log|K_g|=CK_g\] for some real constant \(C\). In particular, the \(\mathcal{E}\)-critical metrics are the elements of \(\mathcal{R}^+_{-2}\). For \(g \in \mathcal{R}^{\pm}_{C}\) and \(\alpha \in \mathbb{R}\) let \(g_{\alpha}=|K_g|^{2\alpha}g\), then \(g_{\alpha}\) has Gauss-curvature \[K_{g_{\alpha}}=(1-C\alpha)|K_g|^{-2\alpha}K_g\] which, for \(\alpha \neq \frac{1}{C}\), satisfies \[\Delta_{g_{\alpha}} \log |K_{g_\alpha}| = \left(\frac{2\alpha -1}{\alpha-\frac{1}{C}}\right)K_{g_\alpha}=C_{\alpha}K_{g_{\alpha}},\] where \[C_{\alpha}=\left(\frac{2\alpha -1}{\alpha-\frac{1}{C}}\right).\] It follows that for \(\alpha>\frac{1}{C}\) the map \(\varphi_{\alpha}\) sending \(g\) to \(g_{\alpha}\) satisfies \[\varphi_{\alpha} : \mathcal{R}^{\pm}_{C} \to \mathcal{R}^{\mp}_{C_{\alpha}}\] whereas for \(\alpha < \frac{1}{C}\) \[\varphi_{\alpha} : \mathcal{R}^{\pm}_{C} \to \mathcal{R}^{\pm}_{C_{\alpha}}.\] Note that the choice \(\alpha=\frac{1}{C}\) maps the elements of \(\mathcal{R}^{\pm}_{C}\) to flat metrics and the choice \(\alpha=\frac{1}{2}\) (assuming \(C\neq 2\)) maps the elements of \(\mathcal{R}^{\pm}_{C}\) to metrics of non-zero constant Gauss curvature.

Suppose \(g \in \mathcal{R}^{-}_{4}\), then \(\hat{g}=g_{3/8}\) is an \(\mathcal{E}\)-critical metric with Gauss curvature \[K_{\hat{g}}=\frac{1}{2}|K_g|^{1/4}.\] It follows with (2.2) and (2.3) that the symmetric trace-free entropy form \[\begin{aligned} T&:=V_{\hat{g}}={}^{\hat{g}}\mathring{\nabla}^2\log K_{\hat{g}}={}^{\hat{g}}\mathring{\nabla}^2\left(\frac{1}{4}\log |K_g|\right)\\&=-{}^{\hat{g}}{\mathring{\nabla}}^2u_g=-\tensor[^g]{\mathring{\nabla}}{}^2u_g-3\left((\mathrm{d}u_g)^2-\frac{1}{2}g(\tensor[^g]{\nabla}{}u_g,\tensor[^g]{\nabla}{}u_g)g\right) \end{aligned}\] is divergence-free with respect to \(\hat{g}\).

If \(M\) is oriented, then we call the associated quadratic differential \(P:=T+iJ T\) the entropy differential. Since the condition on a symmetric \(2\)-form on \(M\) to be trace-free and divergence free is conformally invariant, we obtain:

2.5 The inverse problem

Suppose we are given a Riemann surface \((M,J)\) and a holomorphic quadratic differential \(P\) on \(M\). We ask whether we can locally find a \(J\)-compatible metric \(g\) of negative Gauss curvature on \(M\) which satisfies the Ricci condition and so that the entropy differential of \(g\) is \(P\).

Let \(z: V \to \mathbb{C}\) be local holomorphic coordinates on \((M,J)\). It is easy to check that if the real-valued function \(u\) solves Liouville’s equation \[ 4\partial^2_{z\bar z}u=e^{-2u},\](2.5) then the metric \(g=e^{2u}|\mathrm{d}z|^2\) satisfies the Ricci condition and \(u_g=-\frac{1}{4}\log |K_g|=u\). Now a straightforward computation yields \[\begin{aligned} \operatorname{Re}(P)=T&=-{}^{g_0}\mathring{\nabla}^2u-\mathrm{d}u^2 +\frac{1}{2}g_0({}^{g_0}\nabla u,{}^{g_0}\nabla u)g_0\\ &=-2\operatorname{Re}\left(\left(\partial^2_{zz}u+(\partial_z u)^2\right)\mathrm{d}z^2\right) \end{aligned}\] where \(g_0=|\mathrm{d}z|^2\). Writing \(P=\frac{\rho}{2}\,\mathrm{d}z^2\) for some holomorphic function \(\rho\) on \(V\), we are thus interested in the solutions \(u\) of the system \[4\partial^2_{z\bar z}u=e^{-2u}, \quad \partial^2_{zz} u+(\partial_z u)^2=-\frac{\rho}{4}.\]

We now have the following:

3 Weierstrass Representation

In this section we express the entropy differential \(P\) in terms of the Weierstrass data of the minimal surface \(\Sigma\) – this allows us to compute \(P\) more readily and to easily analyze its singular and asymptotic behavior.

3.1 The Weierstrass Representation

Recall, to an oriented minimal surface \(\Sigma\) in \(\mathbb R^3\) with parametrization \(\mathbf{x}_\Sigma:M \to \Sigma\) one can associate Weierstrass data which encodes the surface and parametrization \(\mathbf{x}_\Sigma\) in complex analytic data. More precisely, the Weierstrass data associated to \(\mathbf{x}_\Sigma\) is the quadruple \((M, J, G, \eta)\) where \((M,J)\) is a Riemann surface, \(G\) is a meromorphic function on \((M,J)\) and \(\eta\) a holomorphic one form on \((M,J)\). The data is determined as follows:

1. \(J\) is the almost-complex structure induced by \(\mathbf{x}_\Sigma\);

2. \(G=S\circ \mathbf{n}\) where \(\mathbf{n}\) is the Gauss map and \[S:\partial B_1 \backslash (0,0,-1)\to \mathbb{C}\] is stereographic projection;

3. \(\mathbf{x}_\Sigma^* \mathrm{d}x_3= \operatorname{Re}\eta\).

The Weierstrass data allows one to reconstruct \(\mathbf{x}_\Sigma\) by the means of the Weierstrass representation: \[ \mathbf{x}_\Sigma(p)-\mathbf{x}_\Sigma(p_0)= \operatorname{Re}\int^p_{p_0} \left( \frac{1}{2} (G^{-1}-G), \frac{i}{2 }( G^{-1}+G) , 1\right) \eta.\](3.1) Conversely, given any quadruple \((M, J, G, \eta)\) we may use [eqn:WeierstrassRep] to construct a parametrization \(\mathbf{x}_\Sigma\) of a branched minimal surface \(\Sigma\) provided:

1. Both \(G \eta\) and \(G^{-1} \eta\) are holomorphic;

2. For any 1-cycle \(\gamma\) in \(M\): \[\int_\gamma \left( \frac{1}{2} (G^{-1}-G), \frac{i}{2 }( G^{-1}+G) , 1\right) \eta\in i \mathbb R^3.\]

Condition [eqn:WRCond2] is known as the period condition.

It is convenient to choose a local complex coordinate patch \((V, z)\) on \(M\) and to write \(\eta=h\mathrm{d}z\) and \(G=G(z)\). We write \(f'\) for \(\partial_z f\) for any function \(f\in C^{1}(V, \mathbb{C})\). Standard computations (see for instance [17]) give the metric as \[g=\mathbf{x}_\Sigma^* g_E= \frac{1}{4} (|G|+|G|^{-1})^2 \eta \otimes \overline{\eta}=\frac{|h|^2}{4} (|G|+|G|^{-1})^2 |\mathrm{d}z|^2,\] the Hopf differential as \[Q = -\frac{1}{G}\mathrm{d}G\circ \eta = -\frac{h G'}{G} \mathrm{d}z^2,\] and the Gauss curvature \[K_{g} =- \frac{16 |G G'|^2}{ |h|^2 (1+|G|^{2})^4}.\] Hence, \[u_g =-\log 2 -\frac{1}{4} \log |h^{-1} G G'|^2 + \log (1+|G|^2).\]

3.2 Computing \(P\) in terms of Weierstrass data

We now compute the entropy differential \(P\) in terms of the Weierstrass data.

As an application of the previous computation, we determine the behavior of the entropy differential at umbilic points of \(\Sigma\):

We may also use Proposition [eqn:WeierstrassCompProp] to compute the entropy differential at branch points.

3.3 Hill’s equation and the (spinor) Weierstrass representation

We conclude by relating the solutions \(w_1,w_2\) from Proposition 2.3 to the Weierstrass data. We observe a connection with the spinorial Weierstrass representation of [18] but do not explore this in depth.

4 Characterization of Minimal Surfaces in Terms of the Entropy Differential

In this section we characterize a number of classical minimal surfaces in terms of the entropy form. In particular, we show that the entropy form vanishes if and only if the surface is contained in Enneper’s surface. The catenoid and helicoid are also characterized in terms of a simple relationship between the entropy form and the second fundamental form.

4.1 Deformed Catenoids and Helicoids

In order to get a complete characterization we must introduce two one-parameter families of surfaces, \(\mathcal{C}\) and \(\mathcal{H}\), which we call, respectively, deformed catenoids and deformed helicoids. Specifically, \(\mathcal{C}\) is the family of surfaces \(C_t\) with Weierstrass data \[\left(\mathbb{C}, J, \frac{t-e^z}{1-t e^z}, \frac{1}{1-t^2} (1-t e^{-z})(1-t e^{z})\mathrm{d}z\right).\] Similarly, \(\mathcal{H}\) is the family of surfaces \(H_t\) with Weierstrass data \[\left(\mathbb{C}, J, \frac{t- e^{z}}{1- t e^{z}}, \frac{-i}{1-t^2} (1-t e^{-z})(1-t e^{z})\mathrm{d}z\right).\] In both cases, \(z\) is the usual coordinate on \(\mathbb{C}\), \(J\) the usual complex structure and \(t\in (-1,1)\). In particular, \(C_0\) is the vertical catenoid and \(H_0\) is the vertical helicoid. Computing as in the preceding section we obtain that for surfaces in \(\mathcal{C}\) \[P=-\frac{1}{2} \mathrm{d}z^2=\frac{1}{2} Q,\] and for surfaces in \(\mathcal{H}\) \[P=-\frac{1}{2} \mathrm{d}z^2=\frac{i}{2} Q.\] We remark that \(\mathcal{C}\) and \(\mathcal{H}\) are obtained from \(C_0\) and from \(H_0\) by applying the one parameter family of Möbius transforms \[B_t:z\mapsto \frac{t+z}{1-t z}\] to the Gauss maps of \(C_0\) and \(H_0\).

Writing \(z=x+iy\) and integrating [eqn:WeierstrassRep] gives the parameterizations of \(C_t \in \mathcal{C}\): \[\begin{aligned} \mathbf{F}_t^{\mathcal{C}}(x,y)&= \mathbf{F}_0^{\mathcal{C}}(x,y)+\frac{2 t}{1-t^2} \left( 0,- y+t \cosh x \sin y , t x-\sinh x \cos y\right) ;\\ \mathbf{F}_0^{\mathcal{C}}(x,y)&=\left(\cosh x \cos y , \cosh x \sin y, x\right). \end{aligned}\] Here \(\mathbf{F}_0^{\mathcal{C}}\) is a parameterization of (an infinite cover of) the catenoid. By inspection, \(C_t\) has \(\Pi_2=\left\{x_2=0\right\}\) and \(\Pi_3=\left\{x_3=0\right\}\) as planes of reflectional symmetry. Moreover, \[\mathbf{F}_t^{\mathcal{C}}(x,y+2\pi)=\mathbf{F}_t^{\mathcal{C}}(x,y)-\frac{4\pi t}{1-t^2} \mathbf{e}_2\] and so \(C_t\) is singly-periodic. When \(t \neq 0\), it is straightforward to see that \(C_t\) is not embedded. Suppose \(E_\theta\) is the rotation of the upper half of \(C_0\) by \(\theta\) around the \(x_2\)-axis. One verifies that \(C_t\) is close to the union of translates of \(E_\theta\) and of \(E_{\pi-\theta}\) where here \(\theta=\tan^{-1}\left( \frac{2t}{1-t^2}\right)\).

Similarly, elements of \(\mathcal{H}\) are parametrized by \[\begin{aligned} \mathbf{F}_t^{\mathcal{H}}(x,y)&= \mathbf{F}_0^{\mathcal{H}}(x,y)+\frac{2t}{1-t^2} \left(0, x+t\sinh x \cos y, t y- \cosh x \sin y\right);\\ \mathbf{F}_0^{\mathcal{H}}(x,y)&= (\sinh x \sin y,- \sinh x \cos y, y). \end{aligned}\] For \(t=0\) this is a parametrization of the helicoid. Note that the image of \(\left\{x=0\right\}\) is a the \(x_3\)-axis while the the image of \(\left\{y=n \pi\right\}\) for \(n\) an integer are the set of parallel lines \(\left\{x_1=0, x_3=\frac{1+t^2}{1-t^2} n \pi\right\}\) contained in the \(\left\{x_1=0\right\}\) plane. Moreover, \[\mathbf{F}_t^{\mathcal{H}}(x,y+2\pi)= \mathbf{F}_t^{\mathcal{H}}(x,y)+2\pi \frac{1-t^2}{1+t^2} \mathbf{e}_3\] so \(H_t\) is singly-periodic. For \(t\neq 0\), \(H_t\) is not embedded. However, if we denote by \(H^\pm_t\) the two components of \(H_t \backslash \left\{x_1=x_2=0\right\}\), then each \(H^\pm_t\) is embedded. In fact, each is a multi-valued graphs over the plane \(\Pi_\theta\) which contains the \(x_2\)-axis and makes an angle \(\theta=\tan^{-1}\left( \frac{2t}{1-t^2}\right)\) with the plane \(\Pi_3=\left\{x_3=0\right\}\). In particular, rotating \(H^\pm_t\) by \(\theta\) around the \(x_2\)-axis gives a surface that looks (roughly) like a sheared copy of \(H_0^\pm\).

4.2 Characterization of minimal surfaces in terms of \(P\) and \(Q\)

We now characterize surfaces in terms of simple relationships between \(P\) and \(Q\). In light of Proposition 2.3, we expect there to be a three-parameter family of surfaces for any fixed of \(P\) and \(Q\). However, in simple settings two of these parameters correspond to re-parameterizations.

5 Curvature Estimates for Embedded Minimal Surfaces in Terms of \(T\)

An interesting problem is to make the characterizations of Theorem 4.1 effective. For instance, to show that a minimal surface with “small” entropy form must be close to a rescaling of a piece of Enneper’s surface. A major challenge is to determine an appropriate notion of smallness for the entropy form – something made more difficult by the need to account for the possible singularities of \(T\). We propose a certain family of quantities as natural ways to measure this smallness and as an application give a novel curvature estimate for embedded minimal surfaces.

Before introducing them we note the following consequence of Corollary 3.2.

We propose that a reasonable notion of size for the entropy differential \(T\) of a smooth minimal surface \(\Sigma\) is given by \[||T||_{\Sigma, \alpha}:= 2^{\frac{1}{2(\alpha+1)}}\int_{\Sigma} |\hat{T}_\alpha|_g^{\frac{1}{1+\alpha}} \mu_g= 2^{\frac{1}{2(\alpha+1)}} \int_{\Sigma} |T|_g^{\frac{1}{1+\alpha}} |K_g|^{\frac{\alpha}{1+\alpha}} \mu_g.\] If \(\Sigma_0\subset \Sigma\), then we obviously have a domain monotonicity property \[||T||_{ {\Sigma_0, \alpha}}\leq ||T||_{{\Sigma, \alpha}}.\] By Lemma 5.1, if \(\Sigma\) is a smooth minimal surface and \(\Sigma_0\) is pre-compact in \(\Sigma\), then \[||T||_{\Sigma_0, \alpha}<\infty\] Finally, if \(\hat{T}^\Sigma_\alpha\) is the \(\alpha\)-weighted entropy form of \(\Sigma\), then \(\hat{T}^{\lambda \Sigma}_\alpha=\lambda^{-2\alpha} \hat{T}^\Sigma_\alpha\) is the \(\alpha\)-weighted entropy form of \(\lambda \Sigma\). To see this observe that \(T\) is scale invariant (by construction) and the Gauss curvature scales like \(\lambda^{-2}\). Hence, as the norm of a (fixed) quadratic differential scales like \(\lambda^{-2}\), \[|\hat{T}_\alpha^{\lambda\Sigma}|_{\lambda g}^{\frac{1}{1+\alpha}} = \lambda^{-2} |\hat{T}^\Sigma_{\alpha}|_g^{\frac{1}{1+\alpha}} \mbox{ and so } ||{T}^{\lambda \Sigma}||_{\lambda \Sigma,\alpha} =||{T}^\Sigma||_{\Sigma, \alpha}\] for all \(\lambda>0\). That is, these quantities are scale invariant for all \(\alpha>0\).

5.1 The Curvature Estimate

We now use the scale invariance of \(||T||_{\Sigma,\alpha}\) and Theorem 4.1 to prove an \(\epsilon\)-regularity result:

We begin with a Lemma which is crucial to the blow-up argument.

We also note the following well-known fact:

5.2 Gap properties of the entropy form

In light of Theorem 5.2 an interesting question is to determine the optimal constant \(\epsilon\) in Theorem 5.2. This is equivalent to determining a lower bound for \(||T||_{\Sigma,\alpha}\) when \(\Sigma\) is a non-flat properly embedded minimal surface in \(\mathbb R^3\). We present some partial results in this direction as well as pose a question about the expected behavior.

A consequence of Theorem 5.2 and [19] is that if \(||T||_{\Sigma,\alpha}\) is finite on a properly embedded surface, then the surface has finite total curvature.

In order to get a more refined result, we first compute:

More generally, we have:

From Proposition 5.7 we obtain two corollaries.

We pose the following question:

5.3 Compactness properties for uniform bounds on \(T\)

We conclude with a compactness result for sequences of properly embedded minimal surfaces \(\Sigma_i\) which admit a uniform bound on the entropy differential. This is a standard consequence of Theorem 5.2 and the removable singularities result of [19].

6 Ricci Solitons

We remark on an interesting connection the entropy form makes between minimal surfaces and two-dimensional Ricci solitons. Recall, a smooth one-parameter family of metrics \(g_t\) on a fixed manifold \(M\) is a Ricci flow provided \[\frac{\mathrm{d}}{\mathrm{d}t} g_t = -2 \operatorname{Ric}_{g_t}.\] This flow was introduced by Hamilton in [13]. When \(M\) is a surface this simplifies to \[\frac{\mathrm{d}}{\mathrm{d}t} g_t =- 2 K_{g_t} g_t.\] We say \((M,g)\) is a Ricci soliton provided there is a vector field \(X\) on \(M\) and a constant \(\lambda\) so that \[-2\operatorname{Ric}_{g}=L_X g-2\lambda g.\] For such a \(g\) the family \(g_t=(1-2\lambda t) \phi_t^* g\) is a Ricci flow – here \(\phi_t\) is the flow of \(X\). When \(\lambda=0\) the soliton is steady (i.e. of unchanging geometry) while when \(\lambda>0\) it is shrinking and when \(\lambda<0\) it is expanding. If \(X=\tensor[^g]{\nabla}{}f\) then we say \(g\) is a gradient Ricci soliton and \(f\) is a soliton potential. For such gradient Ricci solitons \[L_X g= 2\tensor[^g]{\nabla}{}^2 f.\] So \(g\) is a gradient Ricci soliton provided \[\tensor[^g]{\nabla}{}^2 f+ \operatorname{Ric}_{g}-\lambda g=0.\] If \(M\) is a surface this implies \[\tensor[^g]{\nabla}{}^2 f+ K_{g} g -\lambda g=0\] which is equivalent to \[\Delta_gf =2(\lambda-K_g) \quad \text{and}\quad ^{g}\mathring{\nabla}^2 f=0.\] Covariant differentiation of the last equation gives (in coordinates) \[\tensor[^g]{\nabla}{}_j\tensor[^g]{\nabla}{}_k\left(\partial_i f\right)=\frac{1}{2}\left(\partial_k \Delta_g f\right)g_{ij}.\] Using the identity \(\tensor[^g]{\nabla}{}_j\tensor[^g]{\nabla}{}_k\left(\partial_i f\right)-\tensor[^g]{\nabla}{}_k\tensor[^g]{\nabla}{}_j\left(\partial_i f\right)=R^l_{ijk}\partial_l f\), where \(R^l_{ijk}\) are the components of the curvature tensor of \(g\), we obtain \[\left(\frac{1}{2}\partial_k \Delta_g f +K \partial _k f\right)g_{ij}=\left(\frac{1}{2}\partial_j \Delta_g f +K \partial _j f\right)g_{ik}.\] Contracting with \(g^{ij}\) implies \[0=K \mathrm{d}f+\frac{1}{2} \mathrm{d}\Delta_g f=K \mathrm{d}f-\mathrm{d}K.\] Hence near a point \(p\) where \(K_g\neq 0\) we have \[\mathrm{d}f = \mathrm{d}\log |K_g|\] and so \[\tensor[^g]{\nabla}{}^2 \log | K_g|= \tensor[^g]{\nabla}{}^2 f= (\lambda- K_g)g.\] From this we see that \[\Delta_g \log | K_g| =2(\lambda-K_g) \quad \text{and} \quad \mathring{\nabla}_g^2 \log | K_g| =0.\] The converse is also true:

Recall that if \((M,g)\) is a Riemannian surface with \(K_g<0\) and \(g\) satisfies the Ricci condition (2.1), then \(\hat{g}=|K_g|^{3/4} g\) satisfies \(K_{\hat{g}}>0\) and \(\Delta_{\hat{g}} \log K_{\hat{g}}=-2K_{\hat{g}}\). Hence, a consequence of Theorem 4.1 and a straightforward computation is:

References