# Reading roadmap: A Smooth Nonquadratic Entire Affine Maximal Graph\\ in Dimension Ten

The ansatz u(x,t)=sqrt(F(x)^2+t^2) reduces the affine-maximal equation to a radial system. A local integral fixed point supplies a smooth solution at the origin; a box for logarithmic derivatives is then used to continue it globally while retaining positivity.

Source SHA-256: `2cf3fe33debad17053eec4ea32114e45a39290df09f20bcc73898683373c04fe`. Indexed 19,381 characters.

Edges point from prerequisite to consumer. Solid proof edges are attributed inferences, not formal certificates; references remain candidates.

## Internal claim index

### Theorem 1.1 — `thm:main`

Source lines 53–63.

There exists a nonquadratic function $u\in C^\infty(\R^{10})$ with $D^2u$ positive definite at every point such that \[ \sum_{i,j=1}^{10}U^{ij}\partial_{ij} (\det D^2u)^{-11/12}=0 \quad\hbox{on }\R^{10}, \qquad U=\det(D^2u)(D^2u)^{-1}. \]

*Extractive statement; proof gist not independently reconstructed.*

Proof opening: Take the global radial functions of Proposition~\ref{prop:global}. By \eqref{eq:radial}, they satisfy \eqref{eq:reduced} for $x\ne0$. The functions are smooth, $F,P>0$, and $D_x^2F>0$ everywhere, so both identities extend to $x=0$ by continuity. Proposition~\ref{prop:reduction} now shows that \[ u(x,t)=\sqrt{F(|x|)^2+t^2},\qquad (x,t)\in\R^9\times\R, \] is smooth, has positive definite Hessian everywhere, and solves the classical affine maximal equation in dimension ten. Finally, $u(0,t)=\sqrt{1+t^2}$ because $F(0)=1$, so $u$ is not a quadratic polynomial.


Candidate references: `prop:global` (candidate_proof_dependency); `eq:radial` (candidate_proof_dependency); `eq:reduced` (candidate_proof_dependency); `prop:reduction` (candidate_proof_dependency).

### Proposition 2.1 — `prop:reduction`

Source lines 123–138.

Let $F,P$ be positive smooth functions on an open set $\Omega\subset\R^m$, and suppose $H=D_x^2F$ is positive definite. If \det H=FP^{-\gamma},\qquad \tr(H^{-1}D_x^2P)+kP/F=0, then \[ u(x,t)=\sqrt{F(x)^2+t^2} \] has positive definite Hessian on $\Omega\times\R$ and satisfies \eqref{eq:affine} in dimension $n=m+1$.

*Extractive statement; proof gist not independently reconstructed.*

Proof opening: Direct differentiation, interpreted as an identity of quadratic forms, gives D^2u=\frac Fu H+\frac{F^2}{u^3} \left(dt-\frac tF\,dF\right)^2. The first term is positive on the $x$ directions, and the second is positive on the remaining direction. Thus $D^2u>0$. Put $\zeta=t/F$ and use the pointwise basis \[ X_i=e_i+\zeta F_i e_t\quad(1\leq i\leq m),\qquad e_t. \] Its change-of-basis matrix has determinant one. In this basis, \eqref{eq:hessian} has diagonal blocks $(F/u)H$ and $F^2/u^3$, so \det D^2u=F^{m+2}u^{-(m+3)}\det H. The first equation in \eqref{eq:reduced} is equivalent to $P=(F/\det H)^{(m+2)/(m+3)}$. Therefore w=(\det D^2u)^{-(m+2)/(m+3)}=P h(\zeta), \qquad h(\zeta)=(1+\zeta^2)^{k/2


Candidate references: `eq:affine` (internal_reference); `eq:hessian` (candidate_proof_dependency); `eq:reduced` (candidate_proof_dependency); `eq:trace` (candidate_proof_dependency); `eq:det` (candidate_proof_dependency); `eq:affine` (candidate_proof_dependency).

### Lemma 3.1 — `lem:local`

Source lines 220–231.

There is $\varepsilon>0$ and a solution $F,P$ of \eqref{eq:radial} on $0<r<\varepsilon$ such that $F(|x|)$ and $P(|x|)$ extend smoothly to $|x|<\varepsilon$, with \[ F(0)=P(0)=1,\qquad D_x^2F(0)=I,\qquad D_x^2P(0)=-\frac{11}{9}I. \] Moreover $F,P,y,y'>0$ and $P'<0$ for $0<r<\varepsilon$.

*Extractive statement; proof gist not independently reconstructed.*

Proof opening: Seek $F(r)=f(s)$ and $P(r)=p(s)$, where $s=r^2/2$. The integrated equations suggest the normalized derivative $q=y/r$ and flux $j=-y^{m-1}P'/(kr^m)$ for $r>0$. We will construct their regular extensions as functions of $s$ using the following averages. For continuous $f,p$ on $[-\delta,\delta]$ taking values in $[1/2,3/2]$, define q(s)&=\left(m\int_0^1 v^{m-1} f(v^2s)p(v^2s)^{-\gamma}\,dv\right)^{1/m}, \\ j(s)&=\int_0^1 v^{m-1}p(v^2s)^{1-\gamma}\,dv. On the closed subset \[ \mathcal K=\{(f,p)\in C([-\delta,\delta])^2: \|f-1\|_\infty,\|p-1\|_\infty\leq1/2\}, \] consider \mathcal T(f,p)(s)= \left(1+\int_0^s q(a)\,da, \ 1-k\int_0^s j(a)q(a)^{1-m}\,da\right). The functions $q,j$ have uniform pos


Candidate references: `eq:radial` (internal_reference); `eq:q` (candidate_proof_dependency); `eq:j` (candidate_proof_dependency); `eq:fixed-point` (candidate_proof_dependency); `eq:radial` (candidate_proof_dependency).

### Lemma 4.1 — `lem:autonomous`

Source lines 328–341.

On the interval of continuation, $A,B>0$, and the functions $A,B,D$ satisfy \dot A&=A(1+B-A),\\ \dot B&=B\bigl(9+A+(12/11)BD-9B\bigr),\\ \dot D&=(11-D)A-8D+BD(1-D/11). Moreover $(A,B,D)\to(0,1,0)$ as $r\downarrow0$, and all three coordinates are positive for sufficiently small $r>0$.

*Extractive statement; proof gist not independently reconstructed.*

Proof opening: Positivity of $F,y,y'$ gives $A,B>0$. Using \eqref{eq:radial}, one may write \[ B=\frac{r^mFP^{-\gamma}}{y^m}. \] Logarithmic differentiation of $A$ and $B$, followed by the second equation in \eqref{eq:radial}, gives \dot A&=A(1+B-A),\\ \dot B&=B(m+A+\gamma C-mB),\\ \dot C&=C(1+C-(m-1)B)+kAB. For the last identity, use $\dot C=C+C^2-r^2P''/P$ and $AB=r^{m+1}P^{-\gamma}/y^{m-1}$. Thus \[ \dot D=(k-D)A-(m-1)D+BD\bigl(1+(1-\gamma)D\bigr). \] Substituting \eqref{eq:constants} proves \eqref{eq:autonomous}. Lemma~\ref{lem:local} gives $F=1+O(r^2)$, $y=r+O(r^3)$, $y'=1+O(r^2)$, and $P'=-(11/9)r+O(r^3)$. Hence $A\to0$, $B\to1$, and $D\to0$. The positivity of $A,B$ and the local inequality $P'<0$ al


Candidate references: `eq:radial` (candidate_proof_dependency); `eq:constants` (candidate_proof_dependency); `eq:autonomous` (candidate_proof_dependency); `lem:local` (candidate_proof_dependency).

### Lemma 4.2 — `lem:box`

Source lines 378–387.

The box \mathcal B=[0,9/2]\times[0,7/2]\times[0,33/7] is forward invariant for \eqref{eq:autonomous}. The solution arising from Lemma~\ref{lem:local} lies in $\mathcal B$ throughout its interval of continuation.

*Extractive statement; proof gist not independently reconstructed.*

Proof opening: Write $V=(V_1,V_2,V_3)$ for the vector field in \eqref{eq:autonomous}. The upper corner $a_*=(9/2,7/2,33/7)$ satisfies $V(a_*)=0$ by direct substitution. Each $V_i$ is nondecreasing in the other two coordinates on $\mathcal B$. The nonzero cross derivatives are \[ \partial_BV_1=A,\quad \partial_AV_2=B,\quad \partial_DV_2=(12/11)B^2,\quad \partial_AV_3=11-D,\quad \partial_BV_3=D(1-D/11), \] all nonnegative because $0\leq D\leq33/7<11$. On each upper face, comparison with $a_*$ therefore gives the corresponding $V_i\leq0$. On the lower faces $A=0$, $B=0$, and $D=0$, the respective components are $0$, $0$, and $11A$, hence nonnegative. These weak boundary inequalities suffice for invariance. Le


Candidate references: `eq:autonomous` (internal_reference); `lem:local` (internal_reference); `eq:autonomous` (candidate_proof_dependency); `lem:autonomous` (candidate_proof_dependency).

### Proposition 4.3 — `prop:global`

Source lines 429–434.

The solution in Lemma~\ref{lem:local} extends to every $r>0$, with $F,P,y,y'>0$. Its radial extensions to $\R^9$ are smooth, and $D_x^2F$ is positive definite everywhere.

*Extractive statement; proof gist not independently reconstructed.*

Proof opening: Suppose its maximal outer endpoint were $R<\infty$, and fix $r_0\in(0,R)$. Lemma~\ref{lem:box} gives \[ 0\leq A\leq9/2,\qquad 0\leq B\leq7/2, \qquad 0\leq C=BD\leq33/2. \] Since the derivatives of $\log F,\log y,\log P$ with respect to $\log r$ are $A,B,-C$, respectively, integration gives, for $r_0\leq r<R$, F(r_0)&\leq F(r)\leq F(r_0)(R/r_0)^{9/2},\\ y(r_0)&\leq y(r)\leq y(r_0)(R/r_0)^{7/2},\\ P(r_0)(r_0/R)^{33/2}&\leq P(r)\leq P(r_0). Also $|P'(r)|=C(r)P(r)/r\leq(33/2)P(r_0)/r_0$. Thus $(r,F,P,y,Z)$ remains in a compact subset of the domain $r,F,P,y>0$ of the smooth system \eqref{eq:regular}. Its right-hand side is bounded there, so the state has a limit at $R$ in that domain. Local exist


Candidate references: `lem:local` (internal_reference); `lem:box` (candidate_proof_dependency); `eq:regular` (candidate_proof_dependency); `lem:local` (candidate_proof_dependency).

## External inputs and references

### [1] Chern1979 — `external:Chern1979`

Source lines 480–484.

Chern, Shiing-Shen. Affine minimal hypersurfaces. Minimal Submanifolds and Geodesics ({Proceedings of the Japan--United States Seminar, Tokyo, 1977}). 17--30. 1979.


Candidate references: `section:1` (external_reference).

### [2] Calabi1982 — `external:Calabi1982`

Source lines 484–487.

Calabi, Eugenio. Hypersurfaces with maximal affinely invariant area. American Journal of Mathematics. 104. 91--126. 1982.


Candidate references: `section:1` (external_reference).

### [3] TrudingerWang2000 — `external:TrudingerWang2000`

Source lines 487–490.

Trudinger, Neil S. and Wang, Xu-Jia. The {Bernstein} problem for affine maximal hypersurfaces. Inventiones Mathematicae. 140. 399--422. 2000. \url{https://maths-people.anu.edu.au/~wang/publications/1-Invent00.pdf}.


Candidate references: `section:1` (external_reference).

### [4] TrudingerWang2002 — `external:TrudingerWang2002`

Source lines 490–493.

Trudinger, Neil S. and Wang, Xu-Jia. Affine complete locally convex hypersurfaces. Inventiones Mathematicae. 150. 45--60. 2002. \url{https://maths-people.anu.edu.au/~wang/publications/2-Invent02.pdf}.


Candidate references: `section:1` (external_reference).

### [5] Du2024 — `external:Du2024`

Source lines 493–497.

Du, Shi-Zhong. Non-quadratic {Euclidean} complete affine maximal type hypersurfaces for {$\theta\in(0,(N-1)/N]$}. The Journal of Geometric Analysis. 34. 229. 2024. 19 pp..


Candidate references: `section:1` (external_reference).

### [6] SunXingXu2026 — `external:SunXingXu2026`

Source lines 497–501.

Sun, Yalin and Xing, Cheng and Xu, Ruiwei. New non-quadratic {Euclidean} complete affine maximal type hypersurfaces via {Calabi} affine geometry. 2026. Preprint, arXiv:2608.25330. \url{https://arxiv.org/abs/2608.25330}. \url{https://arxiv.org/abs/2608.25330}.


Candidate references: `section:1` (external_reference).

## Coverage and limits

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