Let $M$ be a 3-smooth manifold and $g_{1}$ and $g_{2}$ two conformal metrics on $M$. Consider an immersed surface S in $M$ and let $K_{1}$ and $K_{2}$ be the Gaussian curvatures of $S$ with respect to the induced ambient metrics $g_{1}$ and $g_{2}$.

Question : Has anyone already studied the problem of finding examples of surfaces S such that $K_{1}=K_{2}$?

For the particular case where $M$ is the upper half-space and $g_{1}$ and $g_{2}$ are the usual euclidean and hyperbolic metrics there is a way to find lots of non-trivial examples besides the obvious ones (such as horizontal planes or properly placed spheres).

Can this problem be formulated in terms of some geometric flow? (in the sense that we can start with an intial immersed surface $S_{0}$ and via some evolution in time $S_{t}$ converges to an immersed surface such that $K_{1}=K_{2}$? )

thanks for any help on this!

  • 1
    $\begingroup$ I'm assuming that when you say $g_1$ and $g_2$ are conformal metrics, you mean that there exists a positive function $\lambda$ such that $g_2 = \lambda g_1$. Since Gaussian curvature is an intrinsic invariant of a metric restricted to $S$, your question reduces immediately to the question whether, given two metrics $g_1$ and $g_2$ on $S$ such that $g_2 = \lambda g_1$ for some positive function $\lambda$ on $S$, when is the Gauss curvature $g_1$ equal to the Gauss curvature of $g_2$? The ambient space $M$ plays no real role here. $\endgroup$ – Deane Yang Oct 4 '12 at 1:16
  • $\begingroup$ @Deane, Pedro wants to find a surface in a given ambient space $M$. It is rare thing. It seems that $\lambda\equiv1$ on any such compact surface. $\endgroup$ – Anton Petrunin Oct 4 '12 at 3:38
  • 5
    $\begingroup$ @Anton: Well, that's not quite true. For example, if $\lambda$ is any positive constant other than $1$, then the surfaces in question are exactly the flat surfaces in $M$. Since the $3$-sphere contains the Clifford torus, which is flat and compact, it provides a counterexample. In general, when $\lambda$ is not $1$, the equation that defines these surfaces is a second order Monge-Ampère equation, which can be elliptic, hyperbolic, or degenerate, depending on the sectional curvatures of the two metrics and the derivatives of $\lambda$. There should be many nontrivial examples with $S$ compact. $\endgroup$ – Robert Bryant Oct 4 '12 at 5:00
  • $\begingroup$ Anton and Robert, thanks for clarifying the question for me. Of course, if you start with the surface $S$ and a Riemannian metric on it, and an arbitrary positive function $\lambda$, then it is straightforward to extend both $S$ to an Riemannian 3-manifold $M$ in which $S$ is isometrically embedded and extend $\lambda$ arbitrarily as a positive function on $M$. This gives many examples, including ones where $S$ and/or $M$ are compact. But if you start with $M$ and two metrics on $M$ conformal to each other and ask whether $S$ exists, then indeed that appears to be a much harder question. $\endgroup$ – Deane Yang Oct 4 '12 at 11:41
  • $\begingroup$ Sorry, I am still learning to use mathoverflow, I wanted to add a comment and not an answer... $\endgroup$ – Pedro Roitman Oct 4 '12 at 22:02

I don't know the answers to your questions, i.e., I don't know whether there has been any work already done on this problem, nor do I know whether there is any kind of 'heat flow' argument for constructing solutions.

However, I suspect that the latter, if possible, is not going to be completely straightforward. After all, in the example I mention in my comment above, that of the $3$-sphere with $g_1$ being the standard metric with constant sectional curvature $1$ and $g_2=\lambda g_1$ for some constant $\lambda >1$, the surfaces of interest are the flat surfaces, of which there are many in the $3$-sphere, even compact ones near the Clifford torus. Moreover, because this is a hyperbolic problem in this case, one doesn't have good regularity for the 'stationary' solutions, so finding them by 'heat equation' methods appears to be a doubtful proposition.

In the general case (I'll assume that $M$ is oriented, for simplicity), one can understand the nature of the equations by using the structure equations on the oriented orthonormal frame bundle: Let $\pi:F\to M$ be the orthonormal frame bundle with respect to $g_1$. One has the tautological forms $\omega_i$ such that $\pi^*g_1 = {\omega_1}^2+{\omega_2}^2+{\omega_3}^2$ and the corresponding connection forms $\omega_{ij}=-\omega_{ji}$ satisfying the first structure equations $$ d\omega_i = -\omega_{ij}\wedge \omega_j $$ and the second structure equations $$ d\omega_{ij} = -\omega_{ik}\wedge \omega_{kj} + \Omega_{ij} =-\omega_{ik}\wedge \omega_{kj} + \tfrac12R_{ijkl}\ \omega_k\wedge\omega_l\ . $$

Now, $F$ has a map to the unit sphere bundle $\nu:F\to\Sigma(M)$ defined by sending an orthonormal frame $(p;e_i)$ to the unit vector $(p,e_3)$. The $1$-form $\omega_3$ is the $\nu$-pullback of a well-defined $1$-form on $\Sigma(M)$, which, by abuse of notation, I write as $\omega_3$. Similarly, $\omega_1\wedge\omega_2$ is the $\nu$-pullback of a well-defined $2$-form on $\Sigma(M)$, which I denote by the same symbol. Meanwhile, although $\omega_{12}$ is not the $\nu$-pullback of any $1$-form on $\Sigma(M)$, its exterior derivative $d\omega_{12}= \omega_{31}\wedge\omega_{32} + \Omega_{12}$ is the $\nu$-pullback of a well-defined $2$-form on $\Sigma(M)$.

Now, if $S\subset M$ is any oriented surface in $M^3$, it has a Gauss map $\gamma:S\to \Sigma(M)$ given by sending a point of $S$ to its oriented normal vector. This map satisfies $\gamma^*\omega_3 = 0$, $\gamma^*(\omega_1\wedge\omega_2) = dA$, and $\gamma^*(d\omega_{12})=K\ dA$.

Now, let $g_2 = \lambda g_1$ and set $\lambda = e^{2\mu}$ for some $\mu$. Then the orthonormal frame bundle for $g_2$ can be compared with the orthonormal frame bundle of $g_1$ in the obvious way, and, under that identification, one has $$ \omega^\ast_i = e^\mu\ \omega_i,\qquad\text{and}\qquad \omega^\ast_{ij} = \omega_{ij} +\mu_j\omega_i - \mu_i\omega_j\ , $$ where I have decorated the forms associated to $g_2$ with an asterisk. In particular, one has, computing modulo $\omega_3$ (which is the same as computing modulo $\omega^\ast_3$), $$ d\omega^\ast_{12} \equiv \omega_{31}\wedge\omega_{32} -\mu_3(\omega_{31}\wedge\omega_2 + \omega_1\wedge\omega_{32}) + (e^{2\mu}R^\ast_{1212}+{\mu_3}^2)\ \omega_1\wedge\omega_2\ . $$ The condition that the surface $S$ have the same Gauss curvature with respect to $g_1$ and $g_2$ is then expressed as the condition that its Gauss map $\gamma$ pull back the $2$-form $\Upsilon = d\omega^\ast_{12}-e^{2\mu}d\omega_{12}$ to vanish, since $\gamma^\ast\Upsilon=K^\ast\ dA^\ast - e^{2\mu}K\ dA = e^{2\mu}(K^\ast-K)\ dA$. Now, one computes, using the above formulae, that $$ \Upsilon = (1{-}e^{2\mu})\omega_{31}\wedge\omega_{32} -\mu_3(\omega_{31}\wedge\omega_2{+}\omega_1\wedge\omega_{32}) + \bigl(e^{2\mu}(R^\ast_{1212}{-}R_{1212}){+}{\mu_3}^2\bigr)\omega_1\wedge\omega_2\ . $$

The exterior differential system generated by $\omega_3$, $d\omega_3$ and $\Upsilon$ is a Monge-Ampère system on the $5$-manifold $\Sigma(M)$. Its type depends on the sign of the quantity $$ \Delta = (1-e^{2\mu})(R^\ast_{1212}-R_{1212}) - {\mu_3}^2. $$ Where $\Delta>0$ it is hyperbolic, where $\Delta<0$ it is elliptic, and where $\Delta$ vanishes, it is degenerate.

Note about P. Roitman's example: It is interesting to note that this example is not elliptic everywhere. In fact, using the usual coordinates $x^1,x^2,x^3$ on $M=\mathbb{R}^3$ and letting $u = (u^1,u^2,u^3):\Sigma(M)\to S^2$ be the projection onto the unit $2$-sphere, one has the formulae for the metrics he mentions in the form $$ g_1 = (dx^1)^2+(dx^2)^2+(dx^3)^2\qquad\text{and}\qquad g_2 = \frac{(dx^1)^2+(dx^2)^2+(dx^3)^2}{(x^3)^2}. $$ Then the above formula for $\Delta$ works out to be $$ \Delta = \frac{1-(x^3)^2-(u^3)^2}{(x^3)^2}. $$ In particular, the equation is elliptic in the entire region $x^3 >1$, but in the slab $0 < x^3 < 1$, only the solution surfaces whose normals are 'close enough' to vertical are defined by an elliptic equation. In other words, there are hyperbolic solutions in this 'boundary region'. It might be interesting to see what these non-analytic solutions look like.

Added Comment 1: This example has degenerate solutions (i.e., solutions whose $1$-graphs lie in the locus $\Delta=0$) depending on an arbitrary function of one variable: In fact, one can parametrize them in the form $$ x^1=a(s)+\cos(s)\bigl(t{-}\tanh t \bigr), \quad x^2=b(s)+\sin(s)\bigl(t{-}\tanh t \bigr), \quad x^3 = \text{sech}\ t $$ where $a$ and $b$ are functions of $s$ that satisfy $a'(s)\cos(s)+b'(s)\sin(s)=0$. Choosing $a$ and $b$ to be non-analytic produces non-analytic surfaces.

Added Comment 2: It turns out that there is a Lagrangian $\Lambda_E$ for surfaces of elliptic type (i.e., ones for which $\Delta<0$) in the upper half space such that the elliptic integral surfaces of the above system are extrema of $\Lambda_E$, and there is a Lagrangian $\Lambda_H$ for surfaces of hyperbolic type (i.e., ones for which $\Delta>0$) in the upper half space such that the hyperbolic integral surfaces of the above system are extrema of that $\Lambda_H$. Each of these Lagrangians blows up along the locus $\Delta=0$ in $\Sigma(M)$, and it appears that the degenerate solutions listed in the Comment 1 above are not the extrema of any Lagrangian.

  • $\begingroup$ So it seems to me that given suitable assumptions on $M$ and the metrics $g_1$ and $g_2$ (and therefore on $\mu$) that imply, among other things, the Monge-Ampere system is elliptic, then there really is some hope that $S$ could be found via a heat flow. $\endgroup$ – Deane Yang Oct 4 '12 at 13:35
  • $\begingroup$ @Deane: In the elliptic case it could be, I guess. I don't know of a functional for which these surfaces are critical points, though. That would probably be the right starting point to try to define a flow as a gradient flow of such a functional. $\endgroup$ – Robert Bryant Oct 4 '12 at 15:37
  • $\begingroup$ Robert, I have to say that your analysis makes the situation being studied by Roitman sound like a really interesting geometric/PDE question. It probably would be easiest to focus first on the purely elliptic case. But I agree it would be really cool to find hyperbolic examples, too. Local existence follows immediately but what would be really interesting are global examples. $\endgroup$ – Deane Yang Oct 5 '12 at 12:05
  • $\begingroup$ To all the commentators:Thank you very much for your comments and special thanks to Robert Bryant for his analysis of the problem via EDS. I will proceed with investigations in my very special case (i.e. Upper half space with euclidean and hyperbolic metrics), since in this case there is a way to generate many explicit examples. Surprisingly (for me!) these examples can be generated from holomorphic data (via minimal surfaces (critical poinsts of something!). I'll try to check if this particular case is elliptic and maybe think about some heat flow formulation. $\endgroup$ – Pedro Roitman Oct 10 '12 at 19:33

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