I'm studying by myself Geometric Measure Theory by the book "Sets of Finite Perimeter and Geometric Variational Problems An Introduction to Geometric Measure Theory" written by Francesco Maggi and he assumes that the reader is familiar with the definition of $\mathcal{C}^{1,\gamma}$-hypersurface with $\gamma \in (0,1)$ on the part $\textbf{III}$ of the book, which deals with the regularity theory and analysis of singularities, but I'm not familiar with this definition. I found something about $\mathcal{C}^{1,\gamma}$-maps with $\gamma \in (0,1]$ here and here, but nothing about $\mathcal{C}^{1,\gamma}$-hypersurface with $\gamma \in (0,1)$.
Can someone provide me the definition of $\mathcal{C}^{1,\gamma}$-hypersurface with $\gamma \in (0,1)$? If it is necessary to know previously some properties of these kind of hypersurfaces before continuing my study in GMT, could anyone recommend to me some reference for study these properties?
Thanks in advance!
I would just take that to mean that the hypersurface is a $(d-1)$-dimensional $C^{1,\gamma}$ manifold.
You could thus think of the hypersurface as being locally given by the graph of a $C^{1,\gamma}$ function (perhaps upon reorienting and/or relabeling your coordinate axes).
Edit: I pulled out my copy of Maggi's book and you can look at Thm 26.3 for a precise statement of this regularity theorem (including, of course, the technical details of what is meant by a hypersurface of class $C^{1,\gamma}$). He additionally includes some estimates on the Lipschitz function(s) used to locally characterize the hypersurface.