My primary research interests include differential geometry. In particular, my doctoral research work included Willmore energy and generalized Willmore energy in space forms.
The main goal of my research work is on studying of Generalized Willmore surfaces and flows, by combining methods from pure mathematics (Differential Geometry and PDE) and applied mathematics (finite element method, numerical analysis, computations and modeling of surfaces and flows). I am interested in studying of modeling of surfaces and flows both from a theoretical and from a numerical point of view. My Master thesis was directed by Dr. Magdalena Toda and is entitled “Constant Mean Curvature Surfaces of Revolution versus Willmore Surfaces of Revolution: A Comparative Study with Physical Applications”. This thesis studies some special types of surfaces of revolution and their real world applications. The main two cases hereby considered are the constant mean curvature surfaces of revolution (also called Delaunay surfaces) and Willmores surfaces of revolution, respectively. We first presented some geometric results on Delaunay CMC surfaces which correspond to certain classes of ordinary differential equations and then performed the original construction of Delaunay surfaces, based on roulettes of conics, after which we characterize these geometric objects as solutions to specific ODEs. As a physical application we pre- sented a few physical models of Delaunay surfaces arising as liquid bridges between two vertical walls - which are proved to be unduloidal surfaces, by using Calculus of Variations. We numerically computed the profile curves of these surfaces and provided some numerical models for them. By contrast, we studied Willmore surfaces as minimizers of the Willmore energy (or bending energy).
In particular, we have studied some Willmore surfaces of revolution which come in as solutions to BVP problems consisting of the Willmore equation, together with some special Dirichlet type boundary conditions. We provided some numerical computations for the profiles of these surfaces, using COMSOL Multiphysics. Willmore surfaces of revolution have lots of application in the real world, such as elastic biological membranes. At the microbiological level, a model of such Willmore surfaces of revolution is provided by the beta barrels arising from secondary structures in proteins (beta sheets configured as a rotationally symmetric model).
Articles in journals
. Paragoda T, Application of the moving frame method to deformed Willmore surfaces in space forms. Journal of Geometry and Physics. 128 (2018) 199-208.
. Athukorallage B, Bornia G, Paragoda T, Toda M. Willmore-type energies and Willmore-type surfaces in space forms. JP Journal of Geometry and Topology. 2015;18:93.
. Athukorallage B, Paragoda T, Toda M. Roulettes of conics, Delaunay surfaces and applications. Surveys in Mathematics and Mathematical Sciences. 2014;4:1.
Articles in conference proceedings
. Bhagya A, Aulisa E, Bornia G, Paragoda T, Toda M. NEW ADVANCES IN THE STUDY OF GENERALIZED WILLMORE SURFACES AND FLOW, in Seventeenth International Conference on Geometry, Integrability and Quantization.; 2015:1–11.