# The Morse index of Free Boundary Minimal Hypersurfaces

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A free boundary minimal hypersurface in the unit Euclidean ball is a critical point of the area functional among all hypersurfaces with boundaries in the unit sphere, the boundary of the ball. While regularity and existence aspects of this subjecct have been extensively investigated, little is known about uniqueness. That motivates the study of the Morse index, which quantitatively measures the number of deformations decreasing the area to second order. Henceforth, A. Fraser and R. Schoen proposed a fundamental conjecture concerning surfaces with low indices. In this talk, we discuss recent developments including a joint work with Ari Stern, Detang Zhou, and Graham Smith.