Description and Exploration of Mean-Gauss Surfaces

In this paper we explore solving the prescribed mean curvature equation for surfaces meeting a new relation given by (H_S) = λ(K_S), where H_S and K_S are the mean and Gaussian curvatures, respectively. We prove several existence theorems for various families of surfaces and state a conjecture for surfaces of revolution. To conclude, we state a weak existence theorem, and a strong conjecture concerning possible solutions. The intention is that by using differential geometry tools which would have likely been seen at the undergraduate level, the paper and its results are more accessible. My hope is that these new theorems find applications in the classification of surfaces in the future, or at the very least serves as an interesting curiosity.