Iury Rafael Domingos de Oliveira, Alcides De Carvalho Júnior & Roney Pereira dos Santos
Preface
In 1873, Schläfli posed an enduring natural question about isometric immersions of surfaces that remains unanswered to this day (Yau, 1982; Yau, 1993): can every Riemannian surface be locally isometrically embedded in the flat space R³? A partially affirmative response was achieved in 2003 by Q. Han, J.-X. Hong and C.-S. Lin (Han; Hong; Lin, 2003) contingent upon the behavior of the gradient of the Gauss curvature in the neighborhood of its zeros.
Since this problem is quite general, naturally one asks weaker but geometrically interesting related questions. In 1895, G. Ricci-Curbastro (Ricci-Curbastro, 1895) asked under which conditions a Riemannian surface admits local minimal isometric embedding in R³. As a partial answer, he proved in the same article that at least Riemannian surfaces of strictly negative Gauss curvature K satisfying 4K=Δlog(−K) have this property.
On one hand, by Gauss equation, the squared norm of the second fundamental form α of a minimal surface Σ in R 3 , which is apparently an extrinsic object, indeed satisfies |α|²=−2K. In particular, K is non-positive. On the other hand, the Simons identity (Simons, 1968) states that −2|α|²=Δlog|α|² outside the umbilical points of Σ, and therefore, in terms of the Gauss curvature, we get 4K=Δlog(−K).
Notice that the Simons identity holds even for minimal surfaces of R³ having umbilical points. In this case, it is written as Δ|α|² + 2|α|⁴ − 4|∇|α||² = 0. By replacing ∣α∣ 2 with the Gauss curvature, one obtains the intrinsic equation −KΔK + |∇K|² + 4K³ = 0, which raises the question: by considering K≤0, is this equation enough so that a Riemannian surface whose curvature satisfies it can be minimally realized in R³?
The answer is positive, and it was given in 2015 by A. Moroianu and S. Moroianu (Moroianu; Moroianu, 2015), 120 years after the original question posed by G. Ricci-Curbastro. They showed that a Riemannian surface admits local isometric embedding in R³ as minimal surface when its Gauss curvature K is non-positive and satisfies the intrinsic partial differential equation above.
Since then, the equation −KΔK + |∇K|² + 4K³ = 0 is known as Ricci condition, while the Riemannian surface and its Riemannian metric whose Gauss curvature satisfies −KΔK + |∇K|² + 4K³ = 0 are called respectively Ricci surface and Ricci metric.
The Ricci condition describes the geometry of several known surfaces.
On one hand, any surface with vanishing Gauss curvature is trivially a Ricci surface. But, on the other hand, if we assume that a surface is Ricci and has constant curvature K_0, the Ricci condition implies 4K³_0=0, and so K_0=0.
Every minimal surface in R³ equipped with its induced metric is a Ricci surface with K≤0. Moreover, the maximal space-like surfaces in the Lorentz-Minkowski space L³ equipped with its induced metric is a Ricci surface with K≥0.
Thus, the class of Ricci surfaces is rich and, indeed, larger than the class of minimal surfaces of R³. In fact, some Ricci surfaces are closed and have non-positive curvature, while a minimal surface of the Euclidean space cannot be closed.
This book was prepared for a mini-course at the XXII Escola de Geometria Diferencial, in Teresina, Piauí. It constitutes an introductory text on minimal surfaces of R³ from an intrinsic point of view. We have in mind students with a basic background in Analysis in Rn. So, in Chapter 1 we collect the geometric definitions and tools needed to fix the notation used throughout the text. We recall the definition of smooth and Riemannian surfaces, as well as some of the geometric and analytical objects related to them. We also establish the main concepts about the extrinsic geometry of a surface in R³.
In Chapter 2 we present the main object of study of this book, namely the Ricci surfaces. Our goal is to deduce some useful properties that the Gauss curvature of a Ricci surface must satisfy, and then characterize the minimal surfaces of R³ only by their metrics. As an application, we study some topological obstructions to Ricci surfaces, and we describe the behavior of a warped metric under the Ricci condition.
In Chapter 3 we consider the immersion of a Ricci surface in R³ under geometric constraints typical of the ambient space. First, we classify the Ricci surfaces of R³ that are rotational. Second, we show a relation between curves in R³ having constant torsion and Ricci surfaces in R³ that are ruled. As we will see, the metrics of these two families of surfaces are warped.
In Chapter 4 we state some recent developments and some possible directions to explore in the theory.
