We report our recent work on several questions in geometric and global analysis, with emphasis on the existence and weak stability of isometric immersions and Gauss Codazzi equations. By exploring linkages and analogies between fluid dynamics and differential geometry, we obtain new results and/or counterexamples to these problems. Throughout, the presence or preclusion of ''wild'' spirals and oscillations, reminiscent of the celebrated ''Nash wrinkles'' for $C^ isometric immersions, play a central role in our arguments.

 

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