The governors of New York, New Jersey and Louisiana pointed to tentative signs on Monday that the coronavirus outbreak may be starting to plateau in their states but warned against complacency as the death toll nationwide approached 11,000.

Although coronavirus cases and deaths continued to mount, the governors cited data suggesting the rates of growth and hospitalizations were slowing, possibly signaling a peak was at hand in three U.S. epicenters of the pandemic.

New York Governor Andrew Cuomo said statewide deaths from COVID-19, the highly contagious respiratory illness caused by the virus, were up 599 from Sunday, on par with an increase of 594 during the previous 24 hours and 630 on Friday.

The state’s overall tally of confirmed cases grew by 7% from the previous day to 130,680. But hospitalizations, admissions to intensive care units and the number of patients put on ventilator machines to keep them breathing had all declined, Cuomo said.

“While none of this is good news, the possible flattening of the curve is better than the increases that we have seen,” Cuomo told a daily briefing, referring to the trend line formed when infections, deaths and other data are plotted on a graph. “If we are plateauing, we are plateauing at a high level.”

In neighboring New Jersey, the state with the second-highest number of cases and deaths, Governor Phil Murphy cited a 12% day-to-day growth rate in confirmed positive cases on Monday, half the rate from March 30.

“Our efforts to flatten the curve are starting to pay off,” he said, but warned that a relaxing social-distancing rules would trigger a renewed surge in cases that could yet overwhelm the healthcare system.

Louisiana Governor John Bel Edwards likewise expressed cautious optimism about efforts to contain transmission of the virus in his state, also among the hardest hit.

“New hospital admissions are trending downward,” he told a news conference. “But this will only become a trend if we keep mitigation efforts up.”

The rates of increase also appeared to be easing in Connecticut, where Governor Ned Lamont said, “Our healthcare system is bending but not breaking.”