Is there a way\* to prove that a Bonnet Surface $S$ in isothermal coordinates in $R^3$ with mean curvature ($H$) and Gaussian curvature ($K$) both non-constant and where $(H^2-K)=c$, (with c positive constant), must have negative Gaussian curvature?


Thanks in advance for any help!
Alex


* \*so do not use formulas $A$, $B$ and $P$ in the paper <https://projecteuclid.org/download/pdf_1/euclid.pjm/1102650843>



A kind of surfaces
It is not possible to prove that any surface $S\subset\mathbb{R}^3$ that has both $H$ and $K$ nonconstant but $H^2-K=c^2$ for some constant $c>0$ must have $K$ be negative. In fact, one cannot conclude anything about the sign of $K$ from only these hypotheses, beyond the fact that $K\ge -c^2$. 


In fact, it's not difficult to show, using the Cartan-Kähler Theorem, that there exist plenty of such surfaces with $K$ positive. I can supply the details if you want.


Whether adding the condition that the surface be a Bonnet surface could force $K$ to be negative is another question. I think that is probably not the case, but I'd have to work it out to be sure. 


**Added after the question was changed:** There are *no* Bonnet surfaces $S\subset\mathbb{R}^3$ with the property that $H$ is nonconstant while $H^2-K>0$ is constant (where 'Bonnet surface' means that there is a nontrivial isometric deformation of $S$ in $\mathbb{R}^3$ that preserves the mean curvature $H$). Thus, the OP's question is not sensible. 


A proof that such surfaces do not exist can be got by simply differentiating the equation $H^2-K = r^2$ (where $r>0$ is a constant) several times and combining it with the three equations that follow from requiring that the surface admit a nontrvial $1$-parameter family of isometric embeddings into $\mathbb{R}^3$, all with the same *nonconstant* mean curvature function $H$. 


A careful analysis shows that this overdetermined system is incompatible, even locally. The full analysis is too long to input here. However, you can find a discussion of this in Section 7.2 of [my introductory lectures on exterior differential systems](https://services.math.duke.edu/~bryant/MSRI_Lectures.pdf). If you follow those notes to the discussion on page 50, what this amounts to is seeking a solution to the ODE system on $\mathbb{R}^5$ near the bottom of that page that also satisfies $r\_1=0$ while $H\_1\not=0$ and $H$ and $r$ are nonzero. Since there are no such solutions, there are no such surfaces.


