On the Density Problem in the Parabolic Space

Andrea Merlo, Basque Foundation for Science, Bilbao, Spain. Mihalis Mourgoglou, Basque Foundation for Science, Bilbao, Spain. Carmelo Puliatti, Universidad del Pais Vasco, Leioa, Spain

Chapters; 1. Introduction; 2. Preliminaries and notation; 3. Marstrand's theorem and the structure of uniform measures; 4. Marstrand-Mattila rectifiability criterion; 5. Moments and their expansion; 6. Uniform measures of dimension $n 1$ and quadratic surfaces; 7. Structure of non-degenerated uniform measures; 8. Proof of $1$-codimensional Preiss's theorem; 9. Weak constant density implies bilateral weak geometric lemma; A. Proof of \cref{TH:FAILUREPREISS}; B. Taylor expansion of area on quadratic $t$-cones