Higher order isoperimetric inequalities and rigidity for surfaces in $\R^n$

Abstract

We establish a family of sharp higher order isoperimetric type inequalities for smooth compact surfaces in Euclidean spaces of arbitrary codimension. These inequalities involve arbitrarily high derivatives of the boundary curve, together with the mean curvature vector of the surface. For example, the first and second order inequalities yield the sharp two-sided estimate for the isoperimetric deficit: \begin{align*} & \fracπ{R}\int_{\partial M}|x-Rν|^2\, ds +2π\int_M x\cdot H\, dA \le L^2-4πA\\ \le& \fracπ{R}\int_{\partial M}|x-Rν|^2\, ds +2π\int_M x\cdot H\, dA +\fracπ{3R}\int_{\partial M}|x+R^2κ|^2\, ds, \end{align*} where $R=\frac{L}{2π}$ and the origin is chosen to be the centroid of the boundary. For minimal surfaces, equality in the lower bound holds precisely for flat disks, whereas the second inequality admits non-round equality cases. Our results extend that of Carleman, Sachs, Chakerian, and Li-Schoen-Yau. A distinctive feature of our work is the unexpected rigidity structure arising at higher orders. At the first and second orders, equality curves admit decompositions into circular Fourier modes supported in mutually orthogonal planes. Remarkably, this phenomenon fails at higher orders. We establish a multiplicative factorization of the unit tangent map into elementary planar rotations acting on a fixed unit vector, providing a characterization of equality cases at arbitrary order. As a result, we construct a third order equality curve in $\mathbb R^3$ whose Fourier modes cannot be decomposed into mutually orthogonal circular modes.