Geometric analysis is a vibrant field at the intersection of geometry and analysis, exploring fundamental geometric structures and their analytical properties. This paper provides a comprehensive overview of key theorems, recent advances, and emerging research directions in geometric analysis. It discusses foundational concepts such as the Gauss-Bonnet theorem, Ricci flow, and minimal surfaces, highlighting their applications in mathematics, physics, and computer science. Recent developments in geometric flows, nonlinear PDEs, and future research directions are also explored, emphasizing the field's evolving landscape and potential breakthroughs.

Geometric Analysis, Differential Geometry, Geometric Flows, Ricci Flow, Minimal Surfaces, Nonlinear PDEs, Mathematical Physics, Computer Graphics