Number of singularities of the two-dimensional growth function (Q5952925)

Let \((M,g)\) be a 2-dimensional complete Riemannian manifold. Given a point \(x\) in \(M\) the growth function of \(M\) at \(x\) is the real function \(v\) defined by: \(v(r)=0\) if \(r\leq 0\) and \(v(r)= \text{vol}_g B(x,r)\) if \(r>0\), where \(B(x,r)\) denotes the closed geodesic ball of ray \(r\) around \(x\). Denote by \(R_1(g)\) the set of real number \(r\) such that \(v\) is of class \(C^{1+ \alpha}\) in a neighbourhood of \(r\), for some \(\alpha >1/2\), and by \(R_2(g)\) the set of real number \(r\) such that \(v\) is of class \(C^2\) in a neighborhood of \(r\). The authors prove that for any \((M,g)\) one has \(|\mathbb{R} \setminus R_1(g) |\geq s(M)\) and \(|\mathbb{R} \setminus R_2(g) |\geq S(M)\), where:NEWLINENEWLINENEWLINE\(s(M)=0\) for \(\mathbb{R}^2\) and \(S^2\),NEWLINENEWLINENEWLINE\(s(M)=1\) for \(\mathbb{R}\mathbb{P}^2\), compact surfaces (except \(S^2)\) and for noncompact surfaces of finite type (except \(\mathbb{R}^2)\);NEWLINENEWLINENEWLINE\(S(M) =1\) for \(\mathbb{R}^2\);NEWLINENEWLINENEWLINE\(S(M)=2\) for \(S^2\), \(\mathbb{R}\mathbb{P}^2\) and noncompact surfaces of finite type (except \(\mathbb{R}^2)\);NEWLINENEWLINENEWLINE\(S(M)=3\) for compact surfaces (except \(S^2,\mathbb{R} \mathbb{P}^2)\);NEWLINENEWLINENEWLINE\(s(M)=S(M)= +\infty\) for surfaces of infinite type.NEWLINENEWLINENEWLINEThis means that, except for \(\mathbb{R}\mathbb{P}^2, S(M)\) (respectively \(s(M))\) coincides with the minimum number of critical values (respectively of index 1) of a proper Morse function on \(M\). They also prove that the bounds are sharp. Namely, for any \(M\) and any \(x\) in \(M\), they construct a Riemannian metric \(g\) of constant curvature such that the growth function at \(x\) has as many singularities as the above bounds indicate.