For Holder exponents greater than 1/2, weak C^{1,theta} solutions of the Darboux equation are equivalent to C^{1,theta} isometric immersions, via a new distributional Gaussian curvature and flatness criterion.
Rigidity and flexibility of isometric extensions
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Isometric Immersions and Weak Solutions to the Darboux Equation
For Holder exponents greater than 1/2, weak C^{1,theta} solutions of the Darboux equation are equivalent to C^{1,theta} isometric immersions, via a new distributional Gaussian curvature and flatness criterion.