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The bounded-forcing norm alone cannot control the Hölder seminorm of the Hessian matrix. On a unit Euclidean ball take $u_m=m^{-2}\sin (m{x}_1)$ and $f_m=\Delta u_m=-\sin (m{x}_1)$. The derivative supremum norm $|u_m{|}_2$ and $\Vert f_m{\Vert }_{\infty }$ stay bounded, but the Hessian matrix difference between zero and $\pi /(2m)e_1$ gives $[D^2{u}_m{]}_{\alpha }\ge (2m/\pi {)}^{\alpha }$ on $B_{1/2}$.

A critical point of a twice differentiable function is nondegenerate when its Hessian matrix is invertible there. A positive-definite Hessian gives a strict local minimum, a negative-definite Hessian gives a strict local maximum, and an indefinite Hessian gives a saddle point.

A stationary point $x_0$ of a smooth phase $\varphi$ is nondegenerate when its Hessian matrix at $x_0$ is an invertible matrix. In one dimension, this condition is ${\varphi }^{\prime \prime }(x_0)\ne 0$.

A nondivergence-form elliptic operator applies its principal coefficients directly to the Hessian matrix: $Lu=a^{ij}{D}_{ij}u+b^i{D}_{i}u+cu$. Only the symmetric part of $a$ contributes to the second-derivative term. Uniform ellipticity bounds its quadratic form below by $\lambda |\xi {|}^2$ with a constant $\lambda >0$ independent of position.

A stationary point is nonisolated if every neighbourhood contains other stationary points. For a twice differentiable function, a regular smooth curve of stationary points through a point forces its Hessian matrix to annihilate the curve's nonzero tangent: differentiate the vanishing gradient along the curve. Thus the Hessian matrix has a zero eigenvalue there, but that alone need not determine whether the point is a local minimum or local maximum.

Let $K=-3\log \Gamma$ where $\Gamma >0$ is twice differentiable and homogeneous of degree one. If the Kähler Hessian is invertible, the Euler theorem for homogeneous functions identities give ${\tau }_i{K}_{ij}=3{\Gamma }_j/\Gamma$, then $K_{ij}^{-1}{\Gamma }_j/\Gamma ={\tau }_i/3$ and ${\Gamma }_i{K}_{ij}^{-1}{\Gamma }_j/{\Gamma }^2=1/3$. Multiplication by nine gives the no-scale identity. A positive degree-one $\Gamma$ need not induce an invertible or positive Kähler metric: $\Gamma ={\tau }_1+{\tau }_2$ gives a rank-one metric for $K$. The Hessian matrix $({\Gamma }_{ij})$ itself always has the radial null vector $\tau$, whereas the Hessian matrix of $K$ can be invertible. In a larger theory the relevant full inverse metric, or a decoupled block, must obey the identity.

In an integral $\int a(q)e^{i\lambda \beta (q)}dq$, the real-valued function $\beta$ is the phase. Its stationary points satisfy $\nabla \beta =0$; its Hessian matrix determines the local quadratic oscillation and the signature factor in stationary phase method. Degenerate stationary points and boundaries require separate analysis.